After Bosh stole the ball from Peja Stojakovic with 32 seconds left, James made a high-arching 21-foot jumper over center Jeff Foster just before the shot clock expired.
Rookie Charlie Villanueva followed with a block on Anthony Johnson before James made two free throws to seal the victory.
"Like I told my agent, I'm going to make his job easy. I'm going to do my part during the season. His job is during the offseason," James said.
James -- on his sixth team in five years -- is a candidate for the Most Improved Player award. He averaged 11.6 points and 3.6 assists for Milwaukee and Houston last season, but is averaging 18.9 points and 5.4 assists this season after being acquired in a trade for Rafer Alston.
"My main thing is looking for a home," James said. "The ball is in their court."
James said while his agent handles his business this summer, he'll look to improve by taking up boxing.
"It's only going to help me become a better basketball player," he said. "I'm excited."
"Mike has always been a great player. He's getting a chance now to really show the league that he can carry a team and hit big shots," Jackson said.
Jackson had 26 points and Foster tied a career high with 19 -- all in the first half. Foster, a career backup, dominated in the first half, going 8-for-8 from the field and grabbing 12 of his 16 rebounds.
He went 0-for-2 from the field in the second half.
"They were a little more aggressive," Foster said. "I don't know if they were mixed up in the first half or what they were doing, but my teammates found me. Basically, I had eight dunks, so they obviously figured something out at halftime and took it away."
Toronto trailed by as many as 13 in the third quarter, but Bosh faked a jumper and blew by Foster before throwing down a dunk late in the third, cutting Indiana's lead to two.
Rookie Joey Graham, who had eight points and six rebounds in 26 productive minutes, tied it just before the end of the third with an offensive rebound and layup.
Bosh the gave Toronto its first lead since the first quarter with a dunk and a conversion of a three-point play with 6:40 left.
Toronto outscored Indiana 20-14 in the fourth quarter.
"Jeff played great defense on Mike James on the last shot. It was a tough shot to make and it won the game," Stojakovic said.
Villanueva called his block the "turning point in the game" even though it would have been a tough turnaround shot by Johnson.
Peterson's 3-pointer late in the first quarter set an NBA record for most consecutive games with at least one 3-point field goal.
The previous record of 594 was set by the Miami Heat.
Toronto forward Eric Williams would have started at center, but a sore back sidelined him. ... Foster tied his career high in points previously set Nov. 13, 2003, against Seattle.