Chiasson scored with a wrist shot from the left face-off circle to end an 11-game goal drought.
"I've had good chances in the last six or seven games, it just hasn't been going my way," Chiasson said. "But I found a way tonight and got a big point for our team."
[WATCH: All Oilers vs. Stars highlights]
Mikko Koskinen made 42 saves for the Oilers (36-23-8), which is 8-3-2 in its past 13 games.
"Outstanding," Edmonton forward Ryan Nugent-Hopkins said of Koskinen's performance. "From the first period on, he held us in there when we probably weren't playing our best. On a back-to-back, you need your goalie to step up and he definitely did that."
Video: EDM@DAL: Nugent-Hopkins nets PPG from left circle
The Oilers were coming off an 8-3 win against the Nashville Predators on Monday, when Leon Draisaitl (four goals, one assist) and Connor McDavid (one goal, four assists) each had five points. They remain two points behind the Vegas Golden Knights for first place in the Pacific Division after Vegas defeated the New Jersey Devils 3-0.
The Stars (37-21-8), who have lost three games in a row (0-1-2), trail the Colorado Avalanche by five points for second in the Central Division and lead the Winnipeg Jets by eight for third. Anton Khudobin made 25 saves.
"We did a lot of good things," Dallas forward Corey Perry said. "We were good at 5-on-5. I thought we checked and skated well, and their goalie made some big saves. You have to give him credit."
The Stars outshot the Oilers 16-6 in the first period, but neither team scored.
"That's another first period where we should've come out of there with at least a goal and a lead, and that's a little disappointing not getting it," Dallas coach Rick Bowness said. "But it didn't affect the way we played the rest of the game, which is the most important thing."
Nugent-Hopkins, who was playing his 600th NHL game, scored a power-play goal with a wrist shot from the top of the left circle at 7:24 of the second period to give Edmonton a 1-0 lead.
"We just grinded it out for 60 (minutes)," said Nugent-Hopkins, who had an assist on Chiasson's overtime goal. "Maybe we weren't our best in the first period, but we kept coming. We didn't get down on ourselves, and Mikko obviously held us in, but we just kept pushing and didn't deflate as a team."
The Oilers had 2:46 of consecutive power-play time in the second period, including 1:14 at 5-on-3, after Perry was called for interference against McDavid at 12:52 and Blake Comeau was called for cross-checking Darnell Nurse at 13:38, but the Stars killed each penalty.
"You don't want to give them seam passes and let their guys make plays," Dallas defenseman Andrej Sekera said of the penalty kill. "You have to make sure you either block a shot or make a good play but stay in that position.
"We counted on [Khudobin] to make some big saves for us, and he did a good job of that."
Koskinen made the save on all 31 shots he faced in the first and second periods.
"I think every win gets a lot of confidence for the whole crew, and there is something we can build off of," Koskinen said.
John Klingberg scored on the power play with a one-timer at 6:42 of the third period to tie the game 1-1. The defenseman has scored a goal in two straight games for the first time this season.
"If you look back at this game, this is how we want to play," Klingberg said. "We really had the game the whole 60 minutes. [Khudobin] made the saves when he needed to, but I think we played well today."
The Oilers went 5-for-6 on the penalty kill and 2-for-5 on the power play.
"On special teams, we gave up the one goal, but our power play got a couple we need," Oilers coach Dave Tippett said. "We'll take the points, but we're going to have to be better than that."
Video: EA Sports Overtime Winner: Chiasson lifts Oilers
They said it
"We have to be able to win games like this. As the rest of the year goes on, we can look back at this game. It was a hard-fought game all the way through. This is a playoff atmosphere, and this was a playoff game tonight. The fact that we can get it done tonight is definitely good for the team." -- Oilers forward Ryan Nugent-Hopkins
"It's an emotional game. It's two teams that are battling for points and playoff spots. We knew they played last night and knew that we had to come out and get on the forecheck, get on their defense, and create some havoc. I thought we played well. It was a tough goalie (Mikko Koskinen)." -- Stars forward Corey Perry
Need to know
Nugent-Hopkins has 14 points (five goals, nine assists) in the past eight games. ... Draisaitl had an assist on Nugent-Hopkins' goal and has nine points (six goals, three assists) during a three-game point streak. He leads the NHL with 108 points (43 goals, 65 assists) and is on pace for 134 this season. ... McDavid had an assist on Chiasson's goal and has eight points (one goal, seven assists) during a three-game point streak. ... Klingberg has five points (two goals, three assists) during a four-game point streak. ... Stars forward Jamie Benn had an assist on Klingberg's goal and has seven points (one goal, six assists) during a six-game point streak.
Oilers: At the Chicago Blackhawks on Thursday (8:30 p.m. ET; ESPN+, NBCSCH, SNW, NHL.TV)
Stars: At the Nashville Predators on Thursday (8 p.m. ET; ESPN+, FS-TN, FS-SW, NHL.TV)
Video: Koskinen, Chiasson lead Oilers to overtime victory