Daryl Murphy had given the Tractor Boys the lead only for Sam Vokes to equalise before Campbell secured all three points with just three minutes remaining.
The result gives Mick McCarthy two wins from three since taking the manager’s job at Portman Road while Sean Dyche suffered his first defeat as Burnley boss.
It was a quiet start but Campbell was the first man to threaten for Ipswich as his turn and shot in the box was saved by Lee Grant with Guirane N’Daw putting the rebound over the bar.
Martin Paterson tested Stephen Henderson with a shot from a narrow angle but the goalkeeper made a smart save with his feet at the near post.
Carlos Edwards and Richie Wellens then combined well with Wellens’ final ball finding Campbell in the penalty area but Lee Grant made a smart save to tip his shot around the post.
Grant had his heart in his mouth as he spilled Aaron Cresswell’s cross but the ball was cleared off the line before referee Mark Brown eventually awarded a free kick for a foul on the goalkeeper by Murphy.
But just six minutes into the second half Town took the lead as Grant was not given the benefit of the doubt in a challenge with the Irish striker.
Wellens put a cross into the box and Murphy and Grant went up together with the former Sunderland man heading home.
Grant felt he was fouled and Burnley players surrounded the referee but Brown was unmoved and Murphy was awarded his first strike at Portman Road this season.
Dyche brought on Junior Stanislas and Marvin Bartley as he attempted to get his side back into the game but barring a Stanislas shot over the bar, it was Ipswich who remained in control.
Dyche made his final substitution bringing on Vokes and the striker had an almost instantaneous impact as he levelled with 10 minutes to go.
Stanislas put a cross into the box and Vokes was credited with the final touch as he rose with Danny Higginbotham to meet the ball.
But in the 86th minute substitute Jay-Emmanuel Thomas found Edwards who produced a wonderful cross for Campbell to slot home and secure the win for Town.