Edinburgh's Stuart McInally then touched down under the posts in the second-half, before Dan Parks' boot and a late try from Connacht's Danie Poolman narrowed the gap to a single point.
Parks was eyeing up a drop goal effort in the closing stages and Edinburgh hearts were in mouths with 90 seconds to go - only to see the Scot's effort drift wide, handing Edinburgh the narrow 24-23 victory, a second in a row having seen off the Ospreys last time out.
"We had a very strong side out there and, while we won the match, it could have been Connacht's night," said Bradley.
"It's a desperately difficult place to come and get a win, and the players should be very proud of themselves for doing that, but we need to stop making it so difficult in the future.
"It was a very tight game so we're very happy to get over the line and go home with two wins in a row."
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