Connaught earned their first win at Murrayfield since March 2005 as they outlasted a determined Edinburgh side with out-half Dan Parks scoring 19 points.
Michael Kearney and Robbie Henshaw both scored their first league tries for the province, the latter's 76th-minute try proving crucial as Eric Elwood's men came from behind on three separate occasions in the second half.
The home of Scottish rugby has not been a happy hunting ground for Connaught with just one victory from eight previous visits, so this is a significant result in their push for a best ever finish in the RaboDirect PRO12.
Top scoring out-half Dan Parks converted Henshaw's late try and added a further penalty to rob Edinburgh of a losing bonus point, with the Westerners avenging a 24-23 home defeat from earlier in the campaign.
Connaught are now up to 8th in the table as the campaign reaches its finale.