Patrick Wiercioch had a goal and assist, and Chris Phillips also scored for Ottawa, which won for only the second time in eight games (2-2-4). Turris added an assist, and Ben Bishop made 31 saves in his first start in four games.
The Senators also extended their string of games decided by one goal to 11, which is one short of matching the NHL record set by Chicago during the 1997-98 season.
Turris' goal came on the power play, after Christian Ehrhoff was penalized for interference.
Daniel Alfredsson got the puck at the top of the left circle and faked a shot before sending it low to Turris. Turris one-timed it from a tough angle, and roofed the shot over goalie Ryan Miller's shoulder.
The Sabres did manage to salvage a point in a game they were forced to rally back after blowing a 2-0 first-period lead.
Stafford, a three-time 20-goal scorer, doubled his goal total this season, with his second one tying it with 7:54 left in regulation. Getting to the puck at the left boards, Stafford swung around and directed a shot at the net, which deflected in off the stick of Ottawa centre Zack Smith.
The Sabres enjoyed a two-goal lead going into the second period for the first time in 32 games, dating to March 27 of last year. And they then promptly squandered it.
Ottawa scored twice in the first 10:14, during a stretch in which it out-shot Buffalo 12-2. And then Turris scored the go-ahead goal with 63 seconds left. Turris drove behind the net from the right side and was allowed to come out the other where he avoided Jochen Hecht's weak poke-check, and wrapped the puck in off of goalie Ryan Miller's outstretched skate.
Sabres captain Jason Pominville's lazy giveaway inside his own zone led to Wiercioch's goal 4:03 into the period, when he blasted a shot in from the left point.
Phillips was credited with the next Ottawa goal 6:11 later. Miller stopped Phillips' wobbler from the point, but couldn't control the rebound. Sabres defenceman Jordan Leopold had his back to the play and inadvertently batted the puck in with the back of his glove.
And it could've been worse for Buffalo if not for Miller. During a 10-second span early in the second period, he stopped four shots — two by Jakob Silfverberg and two by Peter Regin — from in close all coming after defenceman Robyn Regehr's lazy giveaway inside his own blue line.
Sabres forward Ville Leino had an immediate impact in his season debut, after missing the first 27 games with a hip injury. He set up Tyler Ennis for a scoring chance 6 1/2 minutes in, and then set up Weber's opening goal 2 minutes later.
Stafford made it 2-0 a little over 6 minutes later during a scramble in front.
NOTES: Sabres RW Patrick Kaleta completed serving his five-game NHL suspension for an illegal hit, and is available to return to play at Washington on Sunday. ... Senators LW Matt Kassian made his NHL season debut after being acquired in a trade with Minnesota on March 12. Kassian had been playing with AHL Houston. ... A day earlier, Sabres GM Darcy Regier said he's approaching the NHL trade deadline April 3 with the mindset of adding what he called "top players," and has no intention to "blow up" the current team, which entered the game 29th in the overall standings. ... Weber's goal was the sixth of his career, and first in 51 games dating to last season. ... Senators D Sergei Gonchar had an assist to extend his point streak to six games.