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.
Both sides were searching for some rhythm in the second period but the action was littered with handling errors.
Leader landed his longest penalty of the night, from 30m, to make it 13-9 and that sparked a period of pressure from the visitors. Again, however, they were unable to make it count.
Edinburgh looked to have given themselves a healthy cushion with 10 minutes left when Du Preez battled to the line following a scrum five metres from the Connacht line, but the TV official decided he had dropped the ball before crossing.
And that was to prove costly as Connacht came storming back.
Replacement Sam Hidalgo Clyne gave away a penalty and after the visitors kicked for touch, they mauled and rolled the ball over the line through Faloon.
The home side piled forward in the last few minutes but their efforts were in vain as Connacht held on for victory.
Tries: Du Preez. Cons: Heathcote. Pens: Heathcote 2.
Cuthbert, Beard, Burleigh, Strauss, McLennan, Heathcote, Kennedy, Dickinson, Ford, Andress, McKenzie, Atkins, Leonardi, Watson, Du Preez.
Dominquez for Cuthbert (54), Hart for Beard (78), Hidalgo-Clyne for Kennedy (53), Hilterbrand for Ford (66), Nel for Andress (40), Bresler for McKenzie (53), Coman for Leonardi (60).
Tries: Faloon. Pens: D. Leader 3.
D. Leader, Adeolokun, Henshaw, McSharry, Poolman, Carty, Marmion, Buckley, Heffernan, White, Kearney, Roux, Muldoon, Faloon, McKeon.
Ronaldson for D. Leader (70), Carr for Poolman (54), Henry for Heffernan (40), Ah You for White (74), Naoupu for Muldoon (54).
Gary Conway (Ireland).