Sean Everitt’s side had to play through two yellow cards and needed the final play of the game to secure the fourth try.
They were in control for most of the afternoon, dominating possession and territory at a ground where wins have been hard to come by in recent years.
James Grayson opened the scoring with an early penalty after Zebre were penalised for sealing off. The momentum didn’t last. Malelili Satala was sent to the sin bin for a deliberate knock-on, and Zebre made the extra man count.
After Ewan Ashman was penalised for not releasing, they took a quick tap. Captain Alessandro Fusco started the move, linked with Simone Gesi and took the return pass to score.
The 14 men nearly hit back straight away. Just before Satala returned, Geordie Gwynn showed real pace and power on the wing and went over in the corner. The TMO found the ball had come loose under pressure from two covering defenders as he grounded it, so the try was ruled out.
Edinburgh kept pressing with their offloading game, then switched to power when they got into the Zebre 22 just before the half-hour.
Big carries from Pierre Schoeman, Ashman and Magnus Bradbury sucked in defenders, and the ball went wide for co-captain Matt Currie, who crashed through two tacklers to put the visitors back in front.
Four minutes before the break, Zebre went down to 14 when Fusco was carded for offside. Edinburgh kicked the penalty to the corner and drove the lineout over, with Freddy Douglas touching down at the back of the maul. That made it 15-5 at half-time.
The start of the second half belonged to Edinburgh. The forwards made wave after wave of carries with penalty advantage, and on 55 minutes co-captain Bradbury powered over under the posts from a Douglas assist. Grayson converted for 22-5. Bradbury had been outstanding all afternoon and fully deserved the try.
Edinburgh went down to 14 again when replacement hooker Harri Morris was shown yellow for a dangerous clear-out. Ashman came back on to cover the front row, and Douglas went off.
Zebre used the period to score their second try just before Morris returned, with Giacomo Ferrari forcing his way over. The conversion made it 22-12.
That left the bonus point on the line, and Edinburgh made sure of it. Once they had the ball back, they kicked a penalty to the corner, and the rolling maul carried replacement back-row Tom Currie over in the final minute. Grayson’s conversion completed a 29-12 win.
The URC now pauses as the European campaign begins. This Friday 16 October (kick-off 8pm) Edinburgh welcome RC Toulon to Hive Stadium for Round 1 of the EPCR Challenge Cup.