The province had won only one of their last 12 league matches - beating the Dragons here at the Sportsground in December - and results have certainly been difficult to come by for Pat Lam's men.
They had to do it the hard way against the eager Scottish side, whose last two league ties were postponed due to bad weather.
Roddy Grant's first try of the season was replied to by a Craig Ronaldson penalty, however the Connacht out-half missed his next three place-kicks as Edinburgh led 7-3 at half-time.
But Michael Swift - Connacht's most-capped player - plunged over for the game's only try in the 47th minute and Dan Parks came off the bench to kick the clinching penalty.
The westerners were unable to capitalise on their early possession and it was the visitors who swept into a seventh minute lead, with Grant driven over following a quick ruck recycle.
Nick De Luca's break up to the hosts' 22 did the initial damage and Grant had good support as he touched down from close range. South African newcomer Carl Bezuidenhout added the conversion.
Kieran Marmion and Robbie Henshaw both threatened for Connacht before they opened their account midway through the half, winning a penalty at an Edinburgh scrum which Ronaldson stuck over.
With a lack of line-breaks and serious try-scoring threats, there was little spark shown by either side approaching the break and Ronaldson's series of missed kicks did little for his or the home fans' confidence levels.
He pulled a difficult effort to the right and wide in the 24th minute, but should have done better when hitting the post on the half hour mark and missing to the left with two minutes remaining.
Edinburgh had the wind at their backs in the second period but it was Connacht who lifted the intensity and accuracy of their play on the resumption.
Eoin Griffin provided the momentum with a rare line break, and Fionn Carr ran a good support line to force Edinburgh to scramble back into their 22.
Cornell du Preez's sin-binning for repeated infringements - five metres out from the Edinburgh line - then gave Connacht the impetus to take the lead.
The Scots' spirited defence finally relented as second row Swift crossed near the uprights after a series of pick and drives. Ronaldson's kicking woes continued though as he hit the post with the conversion.
Bezuidenhout missed a 51st minute penalty and with the wind dying down and their error count rising, Edinburgh's frustrations only grew.
Connacht took control as the second half wore on, with man-of-the-match Denis Buckley a prominent figure up front, and Parks' drilled penalty in the 68th minute was enough to settle the issue.