Edinburgh, by contrast, were still winless coming into the clash but a second-half score from teenager Gregor Turner, coupled with some dead-eye kicking from Greig Laidlaw sealed the win.
And Bradley, who up until 18 months ago was Connacht's head coach, paid tribute to his side's strength of character in finally getting off the mark.
"Ultimately the priority was to get a win and we got that tonight and maintained our unbeaten run against Connacht at home in front of another good crowd at Murrayfield," he said.
"Either side could have won but we got the break with the try to win it - just as they could have done moments later.
"Connacht are always difficult when they have confidence and we needed to repeat the commitment and hunger that we showed against Ospreys and take any opportunities to win this game."
The Gunners' gut-wrenching last-minute defeat to Ospreys last time out had put the dampeners on what had been a man-of-the-match display by Greig Laidlaw.
And the skipper, who switched seamlessly from fly-half to scrum half after the introduction of the try-scorer Turner on Friday night, admitted his side had fought tooth and nail for the win this time around.
"It was not pretty at times but we are off the mark with the win and that is the important thing," he said.
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