Rookie kicker Evan McPherson's 35-yard field goal at the buzzer gave Cincinnati the win after the Bengals went the entire game without a lead.
Burrow completed 25 of 32 passes for 348 yards and a pair of touchdown passes to C.J. Uzomah as the Bengals improved to 3-1.
Lawrence was 17-of-24 for 204 yards as he and first-year head coach Urban Meyer will have to wait at least another week for their first NFL victory after falling to 0-4.
Jacksonville took a 14-0 lead on touchdown runs by James Robinson and Lawrence in the first half, and just missed another TD after they were stopped on fourth-and-goal at the Cincinnati one-yard line just before the break.
The momentum swung the Bengals' way after the break, though, as Burrow hit rookie receiver Ja'Marr Chase for a 44-yard gain, then found Uzomah for a 22-yard touchdown on the opening drive of the second half.
Cincinnati's next two drives also ended in TDs, on a one-yard run by Joe Mixon and a 31-yard Burrow-to-Uzomah strike.
After the Bengals' defence got the ball back with 5:40 to play, Burrow bled out the rest of the clock on a 10-play, 73-yard drive that culminated in McPherson's game-winning field goal.
The top pick in the 2020 NFL Draft, Burrow led a second-half comeback that gave Cincinnati a 24-21 victory over the Jacksonville Jaguars on Thursday.
The Bengals are now 3-1, their best start since 2018, and their second-year quarterback is looking more confident with every game he plays.
He put up his best numbers of the young season Thursday, completing 25 of 32 passes (78.1 per cent) for 348 yards and two touchdowns with no interceptions.
"I think I've gotten better and better each week," Burrow told reporters. "Last week, the defence played so well that I didn't have to do a lot and then tonight, you know, there's going to be games like that where you have to step up and make plays.
"I'm really happy with my performance tonight, I'm just going to keep building on this and keep getting better and better."
After the Bengals went into the locker room at half-time trailing 14-0, all four of their second-half drives ended in points.
The first three resulted in touchdowns – two of them on passes from Burrow to tight end C.J. Uzomah – and the last produced a game-winning field goal by Evan McPherson as time expired.
Burrow was 17-of-20 passing after the break and completed all five of his pass attempts on the crucial final drive.
For a player who missed the end of his rookie season after suffering a serious knee injury last November, the progress has been remarkable.
"Joe is just seeing the field really well right now," said Bengals head coach Zac Taylor. "You just kind of put it in his hands."
That is exactly what Burrow wants, and he is pleased with what he is seeing from everyone around him after four games.
"It's not gonna be perfect," Burrow said. "You're gonna have to find ways to win when everything's against you, your back's against the wall, and that's what we did today and I'm proud of the way we fought."
