Bosh had 31 points and 15 rebounds, Andrea Bargnani scored 28 points and the Raptors beat the Bulls 134-129 Sunday.
Chicago's Ben Gordon had 37 points, including a buzzer-beating jumper over Bosh that sent it to overtime.
"Sometimes you can be like 'Oh my God, woe is me, it went in,'" Bosh said. "You can't panic. We're going to overtime whether we like it or not so we might as well regroup and get it right."
Bosh made sure Toronto got it right, putting the Raptors ahead for good by converting a three-point play with 15 seconds left to play in overtime.
"They came back from an unlikely situation," Bosh said. "They had all the momentum in the world but we withstood it. I'm proud of our guys for showing composure."
Jose Calderon had 22 points and a career-high 19 assists, Shawn Marion scored 16 points and Pops Mensah-Bonsu added 13 as the Raptors set a franchise-high for points and extended their winning streak to a season-high four games. Toronto last won four straight between Dec. 9-14, 2007.
"I wish we'd been playing like this earlier [in the season] but you can't change the past," Bosh said. "All we can do is dictate what we do now. We've just got to keep it going."
Toronto is seven games out of the playoffs with nine left to play.
Chicago rookie Derrick Rose had 23 but the Bulls failed to get their first five-game winning streak since Dec. 11-19, 2006.
"We had fought back so much that we were kind of just going on adrenaline," Gordon said. "In overtime we just weren't able to make the plays to win the game."
Chicago's 112-106 home victory over Indiana on Saturday was its seventh win in eight games, but the Bulls came out flat in this one, trailing by 18 in the first half before rallying in the final quarter.
The Raptors took a 13-point lead into the fourth and were up 103-86 with 8:05 to play before the Bulls began to rally, cutting it to 110-106 with 3:13 left after a 9-0 run.
Bosh converted two free throws with 2:33 to play, ending the Raptors' scoring drought of 2:54, but Kirk Hinrich answered with a 3-pointer to keep the Bulls close.
Toronto looked to have it won when Chicago, trailing by one, came up empty three times on one possession in the final 15 seconds. Gordon missed a jumper, Rose's layup rimmed out and Marion delivered a huge block on Tyrus Thomas to maintain Toronto's slender lead.
"I never, never missed a wide-open layup like that to win the game," a frustrated Rose said.
Calderon and Rose each made a pair of free throws but, on Toronto's next possession, Anthony Parker missed the first of two from the line, giving the Bulls another shot with 6 seconds remaining. Gordon took advantage, draining a jumper over Bosh as the buzzer sounded to send it to overtime tied at 119.
"I knew we had about 5 or 6 seconds so I just wanted to be aggressive," Gordon said. "My intention was to try and get over to the basket but Chris Bosh did a pretty good job of not letting me turn the corner. I just had to pull up and I was able to connect."
Chicago took its first lead of the game with Miller's layup to begin the extra session before Toronto went back in front, 126-125, on Bargnani's 3-pointer with 1:47 remaining. But Calderon had a turnover that led to a layup by Rose, putting the Bulls in front by one with 29 seconds left.
Bosh tipped in his own miss and was fouled by Rose, converting the three-point play to make it 129-127 with 15 seconds to go.
"It really came down to that tip Bosh had," Bulls coach Vinny Del Negro said. "That three-point play was really a big, big play."
Things got worse for Chicago on the next possession when Rose turned it over under Toronto's basket as he tried to find an open teammate.
"I thought I had an opening to pass the ball to Tyrus [Thomas] and one of their guys tipped the ball and I lost it," Rose said. "It's disappointing and that's why I'll put this game on me tonight."
Del Negro chalked it up as a learning experience for Rose, the top pick in last year's draft.
"He tried to make a pass and lost it," Del Negro said. "He's young and he's got to learn. The more we put him in those situations, eventually we'll start making more of those."
Bargnani came up with the loose ball and was fouled, converting both free throws to put Toronto up by four. Gordon made a layup but Calderon sealed it with a three-point play.
John Salmons came in averaging 20.3 points in 19 games with Chicago but scored just 10 in this one. Del Negro said Salmons was hampered by a tight groin.
Calderon's 19 assists matched Damon Stoudamire's franchise-high. ... Bulls guard Lindsey Hunter missed the game with flu-like symptoms. ... Thomas got a technical foul for arguing in the second quarter. ... The Bulls had just three rebounds in the first. ... Buffalo Bills receiver Terrell Owens sat courtside while Toronto Maple Leafs defenseman Luke Schenn watched from a baseline seat.
Isaiah Thomas is starting for the Celtics on Thursday in Game 6 against the Hawks after experiencing no lingering issues after suffering a mild left ankle sprain in Tuesday's Game 5 loss.
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The NBA said Thursday that officials got it right when they didn't call a foul as Dwyane Wade tried to tie the score in the final seconds of Game 5, which the Heat lost to the Hornets 90-88.