Bradley, who was in charge of the Connacht for eight years, was making his first return to Galway since being appointed Edinburgh coach and saw his team trail 10-0 at half-time with Fetu'u Vainikolo crossing.
Edinburgh came back into the game after the break as the RaboDirect PRO12's top try scorer Tim Visser along with Scottish veteran Chris Paterson crossed but three penalties for the hosts along with a last-gasp Mike McCarthy score secured their first league win since September.
Despite reaching the quarter-finals of the Heineken Cup for the second time in their history, Edinburgh's league form has been worrying and they headed into the game off the back of four straight losses in the RaboDirect PRO12.
And it was the home team who made the brighter start with Tiernan O'Halloran punching some early holes in the Gunners' defence.
Edinburgh though started to exert some influence of their own with George Naoupu winning a crucial turnover for Connacht while Gregor Hunter was off target with a penalty.
It took until the 20th minute for the deadlock to be broken when Niall O'Connor slotted a penalty after the visitors strayed offside.
The opening try followed on 32 minutes when debutant Denis Buckley's force in the scrum put Edinburgh on the backfoot and Vainikolo was in the right place at the right time to touch down.
O'Connor converted and although he just pulled an early second-half penalty wide, he was on target moments later to give Connacht a seemingly comfortable 13-0 lead.
But it was anything but comfortable when Gavin Duffy's kick was charged down and Visser showed his predatory instincts to touch down in the 50th minute.
Phil Godman was off target with the conversion and when Tom Brown was yellow carded shortly afterwards O'Connor pulled Connacht 16-5 clear with a penalty.
But back again came Edinburgh as some nice interplay in the backs allowed Paterson to race in with another uncoverted score.
Godman finally found his range with a penalty that took the visitors to within three points on the hour mark.
Matthew Jarvis, on as a replacement for O'Connor, eased some jangling home nerves with a penalty after Edinburgh were again pinged at the scrum on 68 minutes but it still left a tight finale in store.
The majority inside the Sportsground were able to go home happy however, as Connacht controlled possession inside the last couple of minutes and some superb work by the outstanding John Muldoon allowed McCarthy to crash over with Jarvis converting in the final play of the game.
|Connacht Rugby Score Card|
|Edinburgh Rugby Score Card|