The defeat was Edinburgh's first under the three-match tenure of interim coaches Stevie Scott and Duncan Hodge.
Greig Laidlaw banged over eight penalties for the hosts, while his one-time international colleague Dan Parks contributed four penalties and a drop goal, as well as the conversions of tries by Mick Kearney and Robbie Henshaw, and Gavin Duffy weighed in with one penalty.
Parks opened Connacht's account after three minutes with a straightforward penalty from inside the home 22.
Within two minutes, Laidlaw booted an equalising three pointer from a similar range.
The home skipper was on target again 10 minutes later, this time from longer range, after an offence at a scrum 35 metres from the Connacht line.
However, it was scrappy stuff, and there were almost 19 minutes on the clock before either try line came under threat.
The visitors produced a multi-phase effort that was easily repelled by the Edinburgh defence and when the ball was spun wide, the attack floundered on a knock on by Henshaw.
Edinburgh were looking marginally more threatening than the visitors and Tim Visser's first meaningful involvement in proceedings yielded another penalty which Laidlaw again steered between the sticks.
A brief flurry of attacking play by the hosts handed the Edinburgh scrum half a fourth attempt on goal, which he again despatched unerringly to extend his side's lead to nine points after half an hour.
The hosts had appeared to be gaining the upper hand, but the momentum swung in the opposite direction in the 33rd minute when Duffy broke clear and sprinted down the left flank then offloaded to Kearney who trotted in for a try.
Parks added the extra two points but was instantly cast in the role of sinner when he offended on the edge of the Connacht 22.
He earned a yellow card for his troubles and Laidlaw gratefully accepted the opportunity to land his fifth kick of the evening.
However, Connacht trimmed the deficit to two points at the interval when Duffy booted an injury-time penalty.
The hosts raced from the blocks at the start of the second period and Matt Scott came close to unlocking the Connacht defence. In the wake of that effort, Laidlaw registered his first miss of the match.
With his time in the sin bin over, Parks returned to the fray and his first involvement was to bang over a penalty that nudged the Irish outfit into the lead for the first time since the third minute.
With the clock ticking up to 53 minutes, the former Scotland fly half was on target once more.
Edinburgh responded with a rare sortie into the opposition 22. An offence by Ronan Loughney resulted in the Connacht prop being shown a yellow card and Laidlaw booting penalty number six to bring the Scots back to within a point.
Edinburgh's next attack produced a scrum and they shunted the short-handed Connacht pack to earn another penalty which Laidlaw chipped over to leave the home side with a two-point lead entering the final quarter.
The ebb and flow continued and Parks restored a single-point advantage with a neatly taken drop goal, only to see that score cancelled out by an eighth penalty from Laidlaw.
With nine minutes left to play, Parks suffered his only miss and that failure looked to have cost the visitors the match.
But, with three minutes to play, Edinburgh full back Tom Brown fumbled and, in the disarray that followed, Henshaw plunged over for the second Connacht try.
Parks added the conversion then deprived Edinburgh of the narrow defeat bonus point when he thumped over a long range penalty with the final action of the match.