The visitors were firmly in the ascendancy in the opening quarter of Tuesday night's clash at the Ricoh Arena but the Sky Blues took hold from then onwards and were ultimately made to pay for a lack of killer instinct in the final third.
The engaging encounter could well have taken a different path had Swans goalkeeper Dorus de Vries been punished for tugging the shirt of Lukas Jutkiewicz at the start of the second half, with a penalty and possible red card waved away by referee Keith Stroud.
Swansea gave warning of their threat in the 74th minute as Dobbie, Moore and Scott Sinclair combined to hit the woodwork an amazing three times in one move. And Dobbie made no mistake two minutes later as he pounced from 12 yards to round off a well-worked move, leaving the visitors to close the game out and Coventry boss Aidy Boothroyd wondering what might have been.
Early on, Swansea's Darren Pratley overhit a pass with team-mates well placed, Craig Beattie provided a routine save for Keiren Westwood before Sinclair saw his effort deflected wide.
Ashley Williams fired over from Mark Gower's corner as Swansea continued to dictate play.
But Coventry were finally sparked into life by an Aron Gunnarsson effort which forced a parried stop from De Vries.
The chance led to a period of pressure from the hosts in which Marlon King drilled wide and Richard Wood saw his header tipped over the crossbar by Dutchman De Vries.
Boothroyd's side were providing plenty of attacking threat by now and De Vries was called upon to produce a super diving save to push David Bell's 18-yard effort over, before Gunnarsson headed off target from the resulting corner.
The half-time whistle brought a goalless conclusion to an appealing opening 45 minutes, and it took just over 60 second-half seconds for the game's contentious moment.
Sky Blues full-back Richard Keogh sent over a high cross from the right which was fumbled by De Vries, and the Swans keeper then pulled back the shirt of Jutkiewicz with the striker ready to pounce.
Yet, despite huge appeals from all quarters, Stroud may have been unsighted as he somehow waved away the claims.
De Vries was again the scourge of Coventry in the 57th minute as he made a great stop to turn away King's free-kick having initially stepped the wrong way.
Coventry were making all the running after the break, with Swansea struggling to get a foothold on the game.
But they came within inches of taking the lead in the 74th minute as Dobbie hit a post from close range, fellow substitute Moore placed the rebound against the bar before Sinclair headed onto a post at the third time of asking.
Yet Dobbie made no mistake from 12 yards two minutes later, finishing Nathan Dyer's pass to down a Coventry side who deserved a share of the spoils.