Sunderland could only manage to claim a 0-0 draw against Arsenal yesterday.
Sunderland and Arsenal played out a 0-0 draw at the Stadium of Light earlier, in a fixture which was very evenly contested.
Both the Black Cats and the Gunners had chances to win the tie, but they were unable to break the deadlock.
The result has taken Sunderland out of the drop zone, but Allardyce admits that he was slightly disappointed with the way that his side started in the match.
"I am really pleased with the overall performance of all the players,” he said.
"I was slightly disappointed we paid them too much respect first half but we were much better second half.
"We sorted that out at half time, got in their faces more and caused a bit of panic in the Arsenal defence.”
Sunderland flew out of the blocks in the second half, and had their best spell of pressure just before the hour mark.
The likes of Fabio Borini, Lamine Kone and Jermain Defoe all went close in the second period, but a combination of some good saves from Petr Cech and some poor finishing cost them the three points.
The draw has still given Sunderland a huge boost in their battle to beat the drop.
Allardyce’s side are now in 17th place in the Premier League table, with Norwich City slipping into the drop zone.