Whereas, it could prove to be a costly slip-up in the long run for Edinburgh who are still third in the table ahead of their final three matches.
Connacht really should have been ahead at half-time, having opened well and showed some of the form which almost shocked Leinster into submission last time out.
Keatley got them off the mark with a penalty in the third minute and he made a superb break to set up Tuohy for an unconverted eighth-minute try.
Keatley was linking well with the in-form Frank Murphy, behind a cohesive home pack, and he landed a lovely drop goal from distance while Connacht had a penalty advantage.
But try-scoring chances for the hosts were missed when the fleet-footed Fionn Carr decided to kick rather than back his own pace, and Mike McComish and Michael Swift were held up close to the line on separate occasions.
Edinburgh, who had lacked their usual panache, sprung into life in the closing ten minutes of the opening half.
Godman twice kicked penalties to the corner and the pressure told when Dutchman Visser broke through for his eighth try of the league campaign.
Godman added the extras and after a Troy Nathan-led attack came to nothing, Edinburgh clinically countered for their second try, swinging play left and right before Robertson finished well.
Three points ahead at the break, Rob Moffat's men tore into Connacht at the start of the second half and with Jim Hamilton and Alan MacDonald carrying well, they looked poised to score.
Ben Cairns went close to registering a try but Keatley left the Scots reeling, five minutes in, when he intercepted a pass on the ten-metre line and darted over for an encouraging score which he failed to convert.
Edinburgh's all-international pack began to force the issue, particularly at scrum time, and a typically pacy attack from the visitors saw Grant cross for a try which Godman converted.
21-19 up with 20 minutes remaining, Edinburgh were all set to finish the job but Connacht dug their heels in and produced a typically dogged final quarter, as the rain began to teem down.
Chasing their first away win since December, Edinburgh were met with stubborn resistance and as Connacht countered and their forwards built through the phases, they eked out that crucial penalty which Nikora coolly converted.
After the game, Edinburgh coach Rob Moffat expressed his disappointment with his side's overall display but conceded that Connacht deservedly took the four points on offer.
"The bottom line is that Connacht deserved to win the match. We were very poor in the first half and, even though we ended up in the lead at half-time, we didn't deserve to be in front," he said.
"It was a poor performance and that is unacceptable. Connacht played quite well, but basically we were flat tonight.
"We didn't execute the game-plan properly and our punishment was losing the game. We made life hard for ourselves."
|Connacht Rugby Score Card|
|Edinburgh Rugby Score Card|