Burnett won his fourth straight outing and the Blue Jays extended Oakland's losing streak to 10 games, beating the Athletics 6-4 on Thursday night.
"It's big for us to hopefully keep on a good run here because we're far from out of it right now," Burnett said. "That's the team attitude. We want to keep winning ballgames and make it interesting."
Joe Inglett drove in the go-ahead run when he was hit by a pitch with the bases loaded, helping Toronto finish a four-game sweep -- the first over Oakland in the team's 32-year history. The Blue Jays improved to 24-17 under Gaston, who took over when John Gibbons was fired on June 20.
Before the game, general manager J.P. Ricciardi said Gaston will return next season.
Toronto has won 14 of its past 18 home games and, at 59-56, is three games above .500 for the first time under Gaston.
Burnett (14-9) allowed four runs and seven hits in six innings, walking two and striking out six. The right-hander has won eight of 10 starts since Gaston took over.
While he got the win, Burnett wasn't at his best in this one, falling behind on Daric Barton's three-run homer in the second and drawing a visit from Gaston, who sensed Burnett was out of rhythm.
"I like to hear it come from myself rather than tell the pitching coach to go out and do that," Gaston said. "When he goes out there, he's going to say what he needs to say. When I go out there, I'm going to say what I need to say. I thought that was the time to go out there and tell him. His rhythm wasn't quite good but he picked it up and pitched better."
Burnett struck out Ryan Sweeney looking on three pitches after Gaston's visit.
"I wasn't going to give up," Burnett said. "Maybe in the past I'd get a little frustrated and I could have really put fuel on the fire but I've learned a lot to try and deal with things that happen out there and go as far as I can to try and keep this team in line for a W."
Winless since July 27, the A's have dropped seven straight series and are a major league-worst 2-17 since the All-Star break.
"If we do one thing right, we'll do something else wrong," Mark Ellis said. "That's kind of the way it's been going."
The last time Oakland had a losing streak this long was a 12-game skid that began with nine straight losses in September 1995 and continued with three more defeats in April 1996.
Oakland also lost 12 consecutive games in April 1994.
The Athletics have been swept in three straight series of three games or more for the first time in Oakland history. The Kansas City A's had the same run of futility in July 1955.
The lone bright spot for Oakland: rookie reliever Brad Ziegler. He worked a scoreless eighth, extending his streak of shutout innings to 35 -- the longest such run to begin a career in modern major league history (since 1900).
The Blue Jays broke a 4-all tie by scoring twice in the sixth without a hit. Left-hander Jerry Blevins (1-2) hit leadoff batter Adam Lind with a pitch, struck out Kevin Mench and then loaded the bases by walking Scott Rolen and John McDonald.
Blevins said he let the pressure of trying to end Oakland's slump get to him, affecting his control.
"I just kind of seized in the heat of the moment," Blevins said. "It was one of those situations where I tried to do too much."
Down 3-0 on Barton's fifth homer of the year, Toronto came back with four in the fourth, loading the bases with two outs before Rolen and McDonald hit consecutive two-run doubles.
Sweeney tied it at 4 when he led off the fifth with a homer to right.
Justin Duchscherer allowed four runs and seven hits in five innings, raising his major league-best ERA to 2.51. Duchscherer, winless in four starts, walked two and matched a season high by striking out six.
"It just seems right now that things are not going our way," Duchscherer said. "We've got to keep fighting through it until they do."
Toronto CF Vernon Wells, out since July 10 with a strained hamstring, will DH for Triple-A Syracuse when he starts a rehab assignment Friday and could be activated in time for a four-game series at Detroit that begins Monday. ... Oakland OF Jack Cust (thumb) was a late scratch and is day-to-day. He was replaced by Emil Brown. ... Inglett returned to Toronto's lineup after missing the past three games with a sore left knee. ... Mike Torres holds the Oakland record for consecutive scoreless innings with 37 in 1976.