An opening hour bereft of belief and James Henry’s goal for the visitors left Palace at risk of elimination to League One opposition, but two goals from substitute and club-record signing Christian Benteke eventually gave the manager his first win from six games.
They will host Manchester City in the next stage, but more important to Allardyce will have been overseeing his first victory since his appointment and the confidence he will hope they take into the Premier League.
During an uninspiring first half between what appeared two evenly-matched teams, it was Bolton who provided the first attacking threat.
After a run down the left, Zach Clough cut the ball back to Josh Vela and the midfielder sent a powerful shot just wide of the right post.
Palace had little more than probed the visitors’ settled defence with a through-ball from former Bolton winger Lee Chung-yong when Sulley Kaikai drew a foul from Jay Spearing on the edge of the area.
In what perhaps demonstrated the low confidence of the more senior Andros Townsend and Loic Remy, the 21-year-old then took responsibility for the free-kick and came within inches of the top right corner.
By half-time, Joe Ledley had been booked after handling the ball — sending it into the goal — from Kaikai’s cross, and Martin Kelly had had a header saved by Ben Alnwick, but beyond Kaikai’s energy there was little to give the hosts cause for optimism.
It was only three minutes into the second half when Henry — accused of being wasteful in front of goal this season — with little build-up or pressure unexpectedly curled into the top-right corner beyond Julian Speroni.
Palace continued to labour, but 14 minutes later Allardyce, — made a triple substition, replacing Damien Delaney, Ledley and the ineffective Remy with James Tomkins, Jason Puncheon and Benteke.
After a further six minutes, Mathieu Flamini found space on the right wing to cross towards the £27million Benteke, and from within the penalty area the striker produced a trademark powerful header beyond Alnwick and into the back of the net.
Seizing on their momentum, Townsend did similarly on the left, this time sending a low ball towards the 26-year-old, who from a similar position classily controlled the ball before calmly finishing to secure a crucial win 13 minutes from time.
Speroni, Ward, Kelly, Delaney (Tomkins 62 , Fryers, Lee, Flamini, Ledley (Puncheon), Townsend, Remy (Benteke 62), KaiKai.
Alnwick, Wilson, Wheater, Beevers, Andrew Taylor, Spearing, Thorpe, Henry (Taylor 89), Vela, Clough (Clayton 88), Madine (Procter 78).
Stuart Attwell (Warwickshire)