Boston was eliminated from playoff contention when Minnesota beat Baltimore. It's the first time Red Sox have missed the playoffs since 2002.
Toronto has finished third -- behind New York and Boston -- in seven of the past eight seasons.
"You know you don't have anything else to play for so you might as well play for that," Overbay said. "Those guys have been in the top two for 50 years or so. It would be a step in the right direction for us."
Toronto moved within 1½ of Boston for second.
"It would mean a change of the guard," Overbay said. "The next year couple of years hopefully we'll get on top of the division."
Overbay extended his hitting streak to 13 games with his career-high 20th home run, a solo shot off Devern Hansack (0-1) in the fourth. Glaus gave Toronto a 2-1 lead with his 36th homer -- Toronto's 10th back-to-back homer this season.
Dustin Pedroia homered for the Red Sox, who were eliminated from AL East contention on Wednesday.
Manny Ramirez pinch-hit and walked in the seventh -- his first appearance in 13 games. Ramirez has missed 22 of the last 31 games with a sore knee.
Asked if he would start soon, Boston manager Terry Francona said: "I don't think we'll see that tomorrow or the next day, but we'll see it how it goes,"
Francona was ejected in the sixth inning after arguing a call at second base. It was the first ejection for any Red Sox player, coach or manager this season He disagreed that second baseman Pedroia took his foot off the bag before trying to turn a double play.
Second base umpire Doug Eddings ruled Overbay safe at second. Francona jogged from the dugout and argued with Eddings before being tossed.
"He said it was pretty blatant. I didn't really view that way," Francona said.
Burnett (9-8) allowed just two runs and six hits. He struck out eight and walked three.
The Red Sox put the tying run on third with one out in the eighth, but Justin Speier escaped the jam by getting Jason Varitek to fly out to shallow center and Wily Mo Pena to pop up to short.
Vernon Wells hit an RBI triple and Glaus added a sacrifice fly off Mike Burns in the bottom half to give Toronto a 5-2 lead.
B.J. Ryan pitched the ninth for his 35th save in 39 chances, despite giving up an RBI single to David Ortiz that made it 5-3.
Pedroia led of the game with a homer, his second of the season.
Varitek's run-scoring double play tied it at 2 in the sixth, but Overbay's run-scoring grounder gave Toronto the lead in the bottom half.
Hansack allowed three runs and six hits in 5-plus innings in his major league debut.
"I thought he represented himself well," Francona said. "Right from the first pitch of the game it looked like he belonged."
- The game was delayed for a few moments during Ortiz's at-bat in the first by a fire alarm. When it was announced over the PA system they were investigating an alarm in the stadium, Ortiz faked like he was running off, drawing laughter from many.
- The consecutive homers are the first for Toronto since Aug. 18 at Baltimore when Vernon Wells and Glaus hit consecutive homers.