The Bulls evoked memories of their past glory days when they stunned the Chiefs to lead 9-3 at half-time and continued to apply pressure after the resumption.
But with 30 minutes to play, Stevenson received the ball near the 10-metre mark, stepped around his opposite Jamba Ulengo, kicked ahead and won the race for the ball to score in the corner.
It started a three-try, 25-point spree which earned the Chiefs a bonus point to give them 24 points out of a possible 25 from five games.
"Coming off the bye we were eager to start well but it didn't transpire that way. The first 40 was pretty disappointing and when we review it I think there'll be a few hard truths," captain Aaron Cruden said.
"But that second 40 was really for the fans. They came out to see an exciting brand of footy and I think we produced that in the second half."
It was another disappointing finish for the Bulls, who last won in Hamilton in 2010, the same year they last held the Super title.
This year they are now one from five but captain Adriaan Strauss felt they are showing improvement.
"At times we showed some excitement, and some intent but the execution at vital times and pressure moments is not there yet. But we're working on it," he said.
In the first half, the Bulls dominated possession and worked the Chiefs around the ground.
But they were unable to cross the line, and for all the pressure they applied their only points came from three Handre Pollard penalties.
Cruden landed one penalty for the Chiefs but with two other potential shots at goal they opted for the line out drive and lost the ball on both occasions.
The game turned with Stevenson's try in the 50th minute with Cruden's conversion putting the Chiefs ahead for the first time.
Cruden landed a further two penalties and Pollard one to push the Chiefs out to a 16-12 lead before the Chiefs stepped up a gear in the last 10 minutes with tries to Damian McKenzie and Atu Moli.
The Chiefs head to South Africa to play the Stormers next week while the Bulls are in Tokyo to play the Sunwolves for the second time in three weeks.