With several frontline players absent on international duty, it was a Edinburgh side mixed with youth and experience which took to the pitch in freezing conditions.
And Connacht, eager to right the wrongs of last weekend's one-point loss to the Ospreys, seized the initiative early on as they dominated the opening spell.
Edinburgh had not yet paid a visit to opposition territory when Connacht broke the deadlock in the 15th minute.
Out-half Miah Nikora fed Ian Keatley, who came on as an early replacement, and he sent winger Brian Tuohy scampering over in the corner for a five-pointer.
New Zealander Nikora was the source for Connacht's second try just four minutes later, firing a massive cross-field kick into space where a wel-placed Tuohy gathered and dotted down for the simplest of scores.
The game had entered the second quarter when Edinburgh finally ventured beyond the halfway line, the attack yielding a penalty 20 metres out in front of the posts. Chris Paterson duly drilled the ball between the posts for a 10-3 scoreline.
The experienced full-back then drew Edinburgh level after 32 minutes when he took a pass at pace and hurtled through an attempted tackle to race in for the hosts' opening try.
Paterson added the conversion himself to leave the teams locked at 10-10 as they played out the final minutes of the first half.
Crucially, Rob Moffat's men enjoyed a productive spell in the third quarter, taking the lead four minutes into the second period when Paterson landed his second penalty of the night.
Edinburgh extended their lead to six points when - with Paterson being treated for a knock - out-half David Blair was on target with a penalty effort from wide on the left.
The home side boosted their advantage to 11 points when Jim Thompson provided an exquisite pass for fellow winger Tim Visser to sprint in for a 55th minute try - his seventh of the current league campaign.
However, Connacht gritted their teeth in testing conditions and battled back once more. Nikora kicked a penalty before Keatley also landed a three-pointer.
Another Paterson penalty for Edinburgh, whose skipper Roddy Grant had been sin-binned, was cancelled out by Keatley with nine minutes left to play, leaving the score at 24-19.
A Connacht scrum five metres from the home line ensured a tense final few minutes, but the hosts produced a dogged defensive effort to repel a series of attacks and hold on for a valuable four points.
|Edinburgh Rugby Score Card|
|Connacht Rugby Score Card|