But with the visitors reduced to 14 men following Cornell Du Preez's second-half yellow, Swift dotted down to nudge them ahead before Dan Parks' penalty extended their advantage.
It was far from a convincing performance from either side but the hosts did just enough to claim a much-needed victory - just their second of their last 13 RaboDirect PRO12 matches.
Heading into the game, Edinburgh head coach Alan Solomons had admitted his side may be a bit rusty having seen their previous RaboDirect PRO12 game against Newport Gwent Dragons postponed.
But there was certainly no sign of any sluggishness for the Scottish side at the Sportsground as Grant charged over for a try on seven minutes.
Much of it owed to the work of Nick de Luca, who with the ball turned over from Connacht, broke through a gap in their defence before feeding Grant in the corner.
Carl Bezuidenhout, making his first start for Edinburgh, duly added the extras as Edinburgh found themselves 7-0 to the good.
Connacht were struggling to really click into gear as the Edinburgh defence held firm and errors plagued their efforts.
But with their scrum functioning with a degree of efficiency, they were awarded a penalty on 20 minutes which Ronaldson converted from in front of the posts.
It was then Edinburgh's turn to rack up the penalties, pinged for not rolling away and interfering at the line out but Ronaldson missed both penalties either side of the half hour mark.
The fly-half had another chance to pull his side back to within a point before half-time from just inside the touch line but he again failed to split the posts leaving Edinburgh to take a 7-3 lead into the break.
The hosts maintained the pressure after the restart and after winning another penalty almost immediately, Ronaldson opted for a switch in tactics, kicking for touch this time.
Eoin Griffin managed to find the gap from the line out, passing to Fionn Carr but the move eventually lost momentum as Edinburgh turned things over.
But the pressure was not relieved on the Scottish outfit and with another penalty going against them, Du Preez was the man to suffer for repeated infringements, sent to the bin on 45 minutes.
Connacht made the man count almost immediately as following a series of pick and drives, Swift bustled his way over for an 8-7 lead, Ronaldson having missed the conversion.
Bezuidenhout had a chance to hit back but his penalty went just wide before De Luca again made another break for Edinburgh but they were penalised for crossing.
The game was finely balanced going into the final 15 minutes with the home side relieved to see replacement Parks slotting over a penalty on 68 minutes to extend their advantage to four.
And with Edinburgh failed to launch a late fightback, Connacht were left celebrating their first win over Scottish opponents in five attempts.
|Connacht Rugby Score Card|
|Edinburgh Rugby Score Card|