Hallgrimsson made two changes from the comeback victory on Thursday as Ferguson and Jake O'Brien came in for Ryan Manning and Dara O'Shea.
Everton defender O'Brien had the first chance of the game on three minutes after a fast-paced start from the hosts.
He rose highest to meet Robbie Brady's corner and his powerful header was pushed away by Plamen lliev.
The Republic threatened again on 23 minutes when Troy Parrott hit the post from close range after connecting with Brady's low delivery.
As was the case in the first leg, Bulgaria took the lead against the run of play on 30 minutes through a scrappy goal.
From their first corner of the game, Kiril Despodov found Georgi Milanov on the edge of the box and his first-time shot was only half cleared, landing at the feet of Antov who steered home.
Parrott almost levelled but his glancing header from another well-timed Brady cross floated just over the crossbar.
In an almost carbon copy of his first attempt, O'Brien again met a Brady corner, but his weak header was cleared off the line by Simeon Petrov.