West Ham travelled to east Lancashire looking to end the Clarets' streak of 10 home successive victories in this tournament, with Sean Dyche's men further buoyed by eight wins on the spin in all competitions.
However, despite a lacklustre first-half showing, Jack Collison, Stewart Downing and Kevin Nolan were introduced from the bench to turn the contest in the visitors' favour.
They progressed thanks to penalties from Matt Taylor and Collison, the first spot-kick won by Nolan, with Keith Treacy also seeing red for Burnley in stoppage time.
"The introduction of first Jack Collison at half-time then Stewart Downing, then Kevin Nolan as the game went on meant our quality of player got better and our quality of passing got better and better," Allardyce said.
"In the end we were very, very happy that we won and I'm very happy with the team and the effort the players put in."
The Hammers manager, who made nine changes, admitted they were not
at the races in the earlier exchanges, in fact they needed 76 minutes to break the deadlock.
Referee Robert Madley adjudged Nolan to have been fouled by Clarets captain Jason Shackell, who was handed a yellow card for his troubles, although he did appear to have poked the ball away.
"I haven't seen it again because I had too many bodies in front of me, so I haven't looked at the laptop just yet," Allardyce added.
"I'll have a look on the way back.
"I'd like to but because I have to do so much press after the game and I want to get back I can't give you a definitive answer over whether it was or it wasn't. If we've had a bit of luck, we've had a bit of luck."