A tense affair that looked set to be decided by kicks sparked into life in the final 10 minutes and went to the wire before Edinburgh coach Michael Bradley could celebrate success over his former side.
A 14-point contribution from Edinburgh skipper Greig Laidlaw earned him the man of the match award with the stand-off kicking four penalties as well as converting a try by Gregor Hunter.
Connacht relied on the boot of Matthew Jarvis to keep them in the match and his replacement Niall O'Connor accounted for the Irishmen's other points with a penalty and a try.
With eight minutes on the clock, Connacht launched the first attack of any note. Edinburgh number eight Dave Denton ploughed through two tacklers. But, when he turned the ball over in contact, Frank Murphy countered and freed Tiernan O'Halloran who looked set to cross for the opening score. However, home winger Lee Jones made a crunching tackle that drove the Irishman into touch.
Connacht had another opportunity to break the deadlock two minutes later when the hosts offended at the breakdown. But Jarvis was off the mark with his 40-metre penalty attempt.
The visitors continued to apply pressure and O'Halloran again displayed the threat he represented when he charged down an attempted clearance by Tom Brown. Again, Jones came to the rescue for the Scots.
Another failed penalty attempt by Jarvis from an almost identical position and an effort by Laidlaw at the other end that fell short of the target meant the scoreboard was still blank as the clock ticked towards 30 minutes.
Edinburgh could consider themselves lucky to be on level terms, and they celebrated their good fortune after 34 minutes when Laidlaw booted a penalty from 25 metres to nudge them into the lead.
By the interval, the situation had deteriorated for Connacht, with Gavin Duffy yellow-carded and George Naoupu forced to retire through injury. Then, with the momentum firmly in their favour, the hosts earned another penalty and Laidlaw again found the target to double his side's tally.
Within three minutes of the restart, Jarvis had hauled the visitors back into the game when he finally found his range to land a penalty. Laidlaw banged over another kick to restore the six point advantage and, although Jarvis kicked another three pointer, Laidlaw responded in kind once more.
The action continued to flow from end to end and O'Connor - on as a replacement for Jarvis - reduced the leeway with another penalty on 55 minutes.
The disjointed nature of the encounter was not helped by a string of substitutions around the hour mark. But the outcome still hung in the balance.
Just as it appeared that the match would be decided with the boot, the home side grabbed a try. Denton charged down a kick by O'Connor and the ball broke to Hunter who forged ahead and plunged over the line.
Laidlaw's successful conversion extended the advantage to double figures.
Connacht finally sprang to life and, for the first time in the second period, attacked the Edinburgh 22. The recycled ball was worked back to O'Connor who atoned for his earlier error when he darted over for an unconverted score.
And although the visitors finished the stronger of the two sides, Edinburgh held on for the victory.
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