West Ham travelled to east Lancashire looking to end the Clarets' streak of 10 home successive victories in this competition, with Sean Dyche's men further buoyed by eight wins on the spin in all competitions.
"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," said Allardyce.
"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 were in action just 48 hours after drawing with Swansea in south Wales on Sunday and their manager, who made nine changes, admitted they were not at the races in the earlier exchanges.
In fact they needed what Allardyce himself admitted may have been a stroke of fortune to break the deadlock 76 minutes in.
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.
Allardyce added: "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. 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."
There was little doubt in opposing boss Dyche's mind that fortune had favoured the Hammers, though.
"I asked the referee and he said he was happy with his view; I wasn't as happy with his view," added the Clarets boss.
"I think it's a very difficult position to see a penalty from. He's behind the play and three or four players, but I always respect referees' decisions and that's the way it is."
Pushed on whether his skipper also felt hard done by, Dyche continued: "He didn't think there was a lot of contact.
"The referee has a moment to make a decision and I asked the referee about his view. Whether or not it was a penalty is debatable because it's a brush together. There was no major contact, certainly not enough to bring them to the floor, but the referee has to make a decision and he's made one."
Dyche celebrates his one-year anniversary at the Burnley helm on Wednesday yet his celebrations were soured further when Treacy, chasing back to try and stop Collison firing home, tripped the Welsh midfielder in stoppage-time to earn a slightly earlier bath.
"The hard one to take is the fact that, in the modern game, does that really warrant a sending off?" asked Dyche.
"The rule states it does but it is bizarre really. I think the penalty is enough, to be honest. They're the rules, I'm not stating we should change them, I'm just saying it was a nothing incident, it wasn't malicious, he was trying to get back and stop a goal."