Edinburgh (10) 24
Tries: Paterson, Visser Con: Paterson Pens: Paterson 3, D Blair
Connacht (10) 19
Tries: Tuohy 2 Pens: Nikora, Keatley 2
Paterson scored Edinburgh's first try at Murrayfield
Edinburgh extended their winning run at Murrayfield to four matches when they survived a desperate final flourish by a spirited Connacht side.
Brian Tuohy crossed the line twice in the early stages as the Irish side established a deserved early lead.
Chris Paterson responded with a penalty and converted try by half-time.
A Tim Visser try put Edinburgh in control in among kicks from Paterson, David Blair and Connacht's Miah Nikora and Ian Keatley.
The win means Edinburgh lie sixth in the Magners League, with Connacht sitting second from bottom.
With several frontline players absent on international duty, it was an unfamiliar Edinburgh side that took the field in freezing conditions.
And the Irish side sought to capitalise on that as they dominated the early phase of the game.
Visser's second-half try put Edinburgh in control
Edinburgh had not yet paid a visit to opposition territory when Connacht broke the deadlock in 15 minutes.
Keatley - on as a ninth minute replacement - was fed by Nikora and sent Tuohy scampering in at the corner.
Nikora was the source of Connacht's second try three minutes later, firing a massive cross kick into space, where a delighted 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. Paterson duly drilled the ball between the sticks.
Scotland full-back Paterson drew Edinburgh level after 31 minutes when he took a pass at pace and hurtled through an attempted tackle to race in for a score.
Paterson kicked the conversion to leave the teams locked at 10 points apiece as they changed ends.
The Scots took the lead three minutes into the second period when Paterson landed his second penalty of the night.
And the hosts extended their lead to six points when - with Paterson being treated for a knock - Blair was on target with a penalty effort from wide on the left.
Edinburgh boosted their advantage to 11 points when Jim Thompson provided an exquisite pass for Visser to sprint in for a try.
However, Connacht battled back once more and Nikora kicked a penalty before Keatley also landed a three pointer.
Another penalty by Paterson, whose skipper Roddy Grant had been sin binned, was cancelled out by Keatley with 10 minutes left to play.
A Connacht scrum five minutes from the home line ensured a tense final two minutes, but the hosts produced a desperate defensive effort to repel a series of attacks and hold on for the win.
Edinburgh: Paterson, Thompson, Cairns, Houston, Visser, D. Blair, Laidlaw, Traynor, Kelly, Young, Hamilton, MacKenzie, Talei, Grant, McInally.
Replacements: G. Cross for Young (40), S. Turnbull for Hamilton (65), Newlands for Talei (37). Not Used: Gillies, Miven, Samson, Grove, A. Turnbull.
Connacht: Duffy, Monahan, Griffin, Matthews, Tuohy, Nikora, F. Murphy, Wilkinson, D. Murphy, Hagan, Swift, McCarthy, A. Browne, O'Connor, Muldoon.
Replacements: Nathan for Monahan (58), Keatley for Griffin (9), Willis for F. Murphy (64), Rogers for Hagan (55), Upton for A. Browne (68). Not Used: Loughney, Sweeney, McComish.