Connacht left it late against Edinburgh at Murrayfield but Willie Faloon's try made it two wins from two games in the Guinness PRO12.
Following their opening success over Newport, this time they slugged it out to win 14-13 in a poor game that rarely came to life due to a large number of injury stoppages.
Edinburgh took an early lead with a couple of penalties by number 10 Tom Heathcote, before Connacht full-back Darragh Leader levelled up with a pair of kicks of his own.
Edinburgh produced the best rugby of the game at the end of the first half, capped by a try from number eight Cornell Du Preez that Heathcote converted to help the hosts to a 13-6 interval lead.
Leader pulled back three points midway through the second half before in typical Connacht fashion the visitors put in a late burst, and the winning try by Willie Faloon came from a driven line-out five minutes from time.
On duty for his second PRO12 game, in the first match on the brand new Murrayfield pitch, referee Gary Conway awarded two penalties in the opening six minutes.
The first was struck home by Edinburgh fly-half Heathcote, whereas Leader missed his from tight to the touchline.
Heathcote stuck over a second in eight minutes and Leader missed his second kick nine minutes later.
In between, Edinburgh had seized a grip on the game but the closest they came to capitalising was when left winger Nick McLennan could not quite reach a side-footed kick from full-back Jack Cuthbert before it rolled dead.
Profiting from Connacht's first notable spell of possession, Leader got his side on the board with a simple penalty in 23 minutes.
Connacht were then handed a chance when McLennan and Cuthbert messed up a quickly-taken line-out, but from the scrum-five stand-off Jack Carty's prodded kick was touched down by the home side.
However the pressure pendulum had swung the Irish way and when a scrum collapsed after 36 minutes Leader struck the equalising penalty home.
However there was time for Edinburgh to put in some ferocious pick-and-go before a long pass by Sam Beard found number eight Du Preez on the left wing, and he dived over in the corner. The conversion by Heathcote saw Edinburgh go in leading 13-6.
Edinburgh's scrum problems continued into the second half and 15 minutes after the restart, prop WP Nel, who had replaced John Andress at the break, was shown a yellow card at the set-piece and Leader kicked the penalty.
Just as Nel's time-out was ending, Connacht were reduced themselves, when television evidence saw flanker Faloon sin-binned for stamping.
As the game stumbled towards its end, it came alive when Du Preez hoisted a loose ball upfield and isolated Niyi Adeolokun was forced to concede a scrum five. Du Preez was denied a try off the back of it by TV evidence.
Connacht got upfield and put a penalty into the corner from where the pack mauled over with Faloon touching down. Stand-off Jack Carty failed to convert but Connacht had taken the lead, and they clung to it.