Substitute Kyle Hurst broke the deadlock in the 87th minute, sweeping a sure shot into the top corner from the left of the area.
And that sparked a collapse by the Midland League Premier Division side.
A minute later Tommy Anderson’s shot was spilled and Kyle McFarlane was denied on the follow-up by a fine clearing tackle.
But Coventry couldn’t stem the Blues tide as Olly McCoy and McFarlane combined down the right, with the latter crossing low for Caolan Boyd-Munce, who turned the ball onto the base of the post; Hurst was there to collect the loose ball and stick it away (90+1).
Anderson then broke from deep down the middle and shifted the ball to the right for McCoy on the burst, and he dispatched his shot beyond the ‘keeper just inside the post (90+2).
That was that for Blues, whose approach, competitiveness and ultimately those moments of quality ensured they negotiated a tricky First Round tie.
On a bobbly pitch, Coventry played some impressive stuff at times and were full of energy.
Blues made it clear early on that, to a man, they would stand up to the physical task. Some hefty challenges were put in.
The first big opening came in the 28th minute when a deflected shot by Boyd-Munce deflected into the path of McFarlane, who executed a superb turn and got a quick shot away only to be denied by a sharp save from Aaron Stringfellow.
Coventry responded when Charlie Faulkner cracked a fierce half-volley after the ball had dropped from a set-piece and it rebounded off Zach Jeacock’s crossbar.
That was a close shave, as was Demi Smikle sticking wide a low cross from the right from a decent position just before the break.
There was no drop in Blues application after the break and they were first to show when Francis Amartey went at the heart of the Coventry defence and had a goalbound attempt blocked by a defender.
In the 63rd minute Blues made a double change, bringing on Hurst and Geraldo Bajrami; Josh Martin went to right-back and Hurst left wing.
Blues gradually began to knock firmly and regularly on the door.
In the 67th minute, the ‘keeper made an unorthodox but good save with his feet, thwarting an effort that was angling towards the bottom corner after Anderson played a ball over the top for McFarlane to latch onto.
With 20 minutes to go a move from right to left saw Boyd-Munce set up Hurst but he hoisted over the crossbar from a great position.
The action swung from end to end and substitute Lewis Hudson had a good sight of Blues goal and let fly, Jeacock saving well as the ball dipped awkwardly in front of him.
Then came the pivotal moment of Hurst and Blues’ opening goal, caused when Joe Redmond broke up an attack, strode forward down the centre of the park and kept on going before passing to his left. Hurst checked onto his right foot and produced an accomplished finish into the top corner.
That broke the Coventry resistance and the foundation for the devastating late goal salvo was built not least on a solid, purposeful attitude and a competitive defensive line throughout.
Blues: Jeacock, Dawes (Bajrami 63), Forrest, Redmond, Martin, McCoy, Stirk (McGilp 69), Boyd-Munce, Amartey (Hurst 63), McFarlane, Anderson. Sub not used: Bradley-Hurst.