Bradley enjoyed a successful spell as director of rugby at the Irish outfit before going on to join Edinburgh in May of last year.
His new charges made a bright start but Gregor Hunter was wide with a 14th-minute penalty attempt.
Niall O'Connor was successful with his first shot at goal, getting the scoring underway for Connacht in the 20th minute after Edinburgh were found guilty for being offside.
Connacht then grabbed the only try of the first half with eight minutes to go before the break after an almighty shove by the pack on Edinburgh's put in at the scrum.
Scrum-half Frank Murphy reacted quickest when the visitors' back row failed to control possession, and his turnover set up the move which led to Tongan winger Fetu'u Vainikolo's maiden try. O'Connor's trusty boot supplied the extras.
Godman was introduced by Bradley at half-time for the less experienced Hunter and Edinburgh's control improved as a result.
O'Connor's long-range penalty made Edinburgh's task more difficult, but they were gifted a foothold in the game when Gavin Duffy's clearance kick was charged down to let Tim Visser in for his eleventh try of the season.
Tom Brown was yellow-carded before veteran Chris Paterson doubled 14-man Edinburgh's tally with their second try of the night. Godman, who missed both conversion attempts, struck a penalty to reduce the arrears further and set up a tense finish.
Replacement Matthew Jarvis restored Connacht's lead to six and a clean sweep of Irish wins on the night was sealed by Mike McCarthy's try in the corner, which Jarvis converted.