The Seattle Kraken won the No. 2 pick, and the Anaheim Ducks have the No. 3 pick.
The Sabres, who at 16.6 percent had the best chance of winning the lottery, would pick No. 1 for the fourth time, the first since selecting defenseman Rasmus Dahlin in the 2018 NHL Draft.
"It's a great moment for our franchise," Buffalo general manager Kevyn Adams said. "It's obviously been a tough year for a lot of different reasons, and we're here. We're very excited. We're proud to have this selection, and it's a step in the right direction, and I got a text a couple of minutes ago from our owner saying, 'Smile,' so now I can smile."
The first round of the draft will be held July 23 and rounds 2-7 are July 24.
The lottery included the 15 teams that failed to qualify for the 2021 Stanley Cup Playoffs and the expansion Kraken, who begin play next season.
The odds for the 15 existing teams were based on the inverse order of the regular-season standings. To accommodate the addition of Seattle, the odds of those 15 teams were reduced proportionally from the odds used in the 2020 NHL Draft Lottery.
Video: Gary Bettman conducts 2021 NHL Draft Lottery
The Kraken entered the lottery with the third-best odds, as the Vegas Golden Knights did in their inaugural draft lottery in 2017. Seattle had the same odds (10.3 percent) as the New Jersey Devils, who had the third-worst regular-season record and will pick No. 4.
"We're very happy about this outcome," Seattle general manager Ron Francis said. "... I think certainly when you build your team, you build it in the middle. So that's goaltending ... defense and centermen. But we're open to looking at who the best player is going to be."
Seattle moved ahead of Anaheim, which had the second best odds (12.1 percent) and has never picked No. 1 in an NHL Draft.
"It is obviously disappointing for us and our fans to not remain in the top two of this draft, but we still have the opportunity to make a top five selection for the first time in 16 years," Anaheim general manager Bob Murray said. "The top part of the draft has some outstanding talent, and we will look to add to the exciting young prospects in our organization for years to come."
Unlike previous years, only the top two picks were determined by the lottery, and teams were limited in how far they could rise or fall, based on their regular-season standing. This is the third time in the past 11 draft lotteries that the team with the worst record received the No. 1 pick.
Picks 17-32 will be determined by the results of the Stanley Cup Playoffs. The Arizona Coyotes were selected to pick at No. 11, but they must forfeit their 2021 first-round pick for violating the NHL Combine Testing Policy during the 2019-20 season.
The Sabres this season set an NHL shootout-era record (since 2005-06) of 18 games without a win (0-15-3). Buffalo has not made the playoffs for 10 straight seasons, which is tied for the longest drought in NHL history.
Center Jack Eichel scored two goals in 21 games before his season was ended by a neck injury, and coach Ralph Krueger was fired March 17 and replaced by Don Granato.
"It was a challenging season, a lot of adversity," Adams said. "We were in the position we were in, and you certainly don't want to be there, but that's the reality. So then what comes out of it, knowing that, you know the odds were the highest that you end up [No. 3], you kind of understand that going in. So to have this fall our way, to be in this position, it makes some of the things we went through this year, feel a little bit better about it. But it's a huge opportunity for our franchise to improve. I feel like this is a really big step, something we needed and are excited about."
Seattle will participate in the 2021 NHL Expansion Draft to acquire players two days before taking part in its first NHL Draft.
"The expansion draft is for now," Francis said. "That's the lineup we're trying to field when the season starts in October. The amateur draft, you look at it with one eye for now, obviously … but the big priority is making sure that you draft the right guys and then you put them in a proper environment and let them develop and learn so that they have the best opportunity possible to play in the National Hockey League for your franchise."
[Draft Lottery Lookup Tables: By Team Sequence | By Combination ID]
The Sabres could select defenseman Owen Power, who was No. 1 in NHL Central Scouting's final ranking of North American skaters. He scored 16 points (three goals, 13 assists) and was plus-18 in 26 games this season for the University of Michigan and was named to the Big Ten All-Freshman team.
"I have watched him," Adams said. "I certainly will be watching a lot more of him and others here in the coming weeks. Obviously, he's had a good year in Michigan and he's played some big minutes over on Canada in the [IIHF] World Championship.
"What's exciting about this draft, I think there's a number of players to really watch and dig into, and I'm looking forward to those upcoming meetings with our staff and to really debate. I really enjoy the amateur process, I enjoyed the draft last year, enjoyed kind of going through it and projecting out players and kind of challenging scouts to what they see and what they like or what they don't like. So I'm looking forward to doing that over the coming weeks, and it's nice to have No. 1 where you're doing that and knowing that you can take the player you ultimately land on."
William Eklund, who was No. 1 in NHL Central Scouting's final ranking of International skaters, is another option after the left wing scored 23 points (11 goals, 12 assists) with Djurgarden of the Swedish Hockey League.
Two other top defenseman prospects are Simon Edvinsson, No. 2 in Central Scouting's final ranking of International skaters, and Luke Hughes, No. 4 in Central Scouting's final ranking of North American skaters.
"We need to improve in every area, so we're certainly not going to focus on a position or a need," Adams said. "We're looking to definitely take the best player and the player that we project out to become ... what are they going to be like at 22-23 years old as well."
Edvinsson scored six points (one goal, five assists) in 14 games for Frolunda in Sweden's junior league. He had one assist in 10 games with Frolunda in the SHL, and scored four points (one goal, three assists) in seven games for third-place Sweden at the 2021 IIHF Under-18 World Championship.
Hughes scored 34 points (six goals, 28 assists) in 38 games for the USA Hockey National Team Development Program Under-18 team of the United States Hockey League. He is the younger brother of Vancouver Canucks defenseman Quinn Hughes and Devils forward Jack Hughes, and is committed to play at Michigan next season. The 17-year-old has resumed skating after he had a lacerated tendon in his foot from a skate cut March 7.
"I'm sure if you asked our players, they're excited right now," Adams said. "It's energizing to our fan base, it's energizing just to the people that work in this organization that love this place so much. It's all the above, and it's a big step in the right direction of where we're headed, and we're excited to add this young next great Buffalo Sabres player to our franchise."
2021 NHL Draft order
1. Buffalo Sabres
2. Seattle Kraken
3. Anaheim Ducks
4. New Jersey Devils
5. Columbus Blue Jackets
6. Detroit Red Wings
7. San Jose Sharks
8. Los Angeles Kings
9. Vancouver Canucks
10. Ottawa Senators
11. Arizona Coyotes - forfeited first-round pick
12. Chicago Blackhawks
13. Calgary Flames
14. Philadelphia Flyers
15. Dallas Stars
16. New York Rangers
Picks 17-32 will be determined by the results of the Stanley Cup Playoffs.
NHL.com independent correspondents Andrew Eide and Heather Engel contributed to this report