The Black Cats created more clear-cut chances than the Gunners but were let down by a lack of cutting edge in front of goal as they slipped to a 3-1 defeat.
Even when they did find the net it was courtesy of an Olivier Giroud own goal which cancelled out Joel Campbell's opener for the hosts.
Giroud then made amends by heading Arsenal back in front after the break before Aaron Ramsey added the gloss with a stoppage-time effort.
The loss comes on the back of successive victories that saw Sunderland haul themselves out of the relegation zone - but Allardyce believes he saw enough from his players to have warranted a third straight win.
"We could have won, never mind a draw," he said.
"But we didn't because of our frailties in front of goal - we didn't score when we created chances to get in front and ended up losing and when we had a chance to get a last-gasp equaliser, which we deserved.
"I can honestly say I've been here with Bolton, Blackburn, West Ham and Newcastle but never been here and created as many chances as we created today that's for sure - but the sad thing is we have still lost 3-1.
Allardyce has now been in charge since October 9 and has led Sunderland to three wins in six league games - but felt this was the best he had seen in that time.
"I'm encouraged by the player's performances and I'm really sad for them they didn't have their shooting boots on," he added.
"We have proven we can score goals and could've scored a few today but unfortunately we didn't
We failed to get a result and it is our fault - nobody else's.
"It was a good performance; I can't knock the players for their efforts
It's the best performance I've seen since I've been at the club.
"The players are improving, each week they're performing better and better, it's just unfortunate that it didn't get us a result today and I think we deserved to."
Arsene Wenger was satisfied to end a run of three Premier League games without a win as Arsenal moved up to second in the table and he praised his side's response to conceding an unfortunate equaliser on the stroke of half-time.
"It was a very important game for us tonight because we came out of three disappointing results," he said.
"We had lost only one game but I think it was a very, very important game for us today, psychologically, for the fact we had not won, for the fact we had lost important players in our side and we had to show that we can win.
"All was not perfect of course but the team have shown mental strength after a goal before half-time
We still managed to win the game in the second half."
Ahead of their pivotal Champions League clash against Olympiacos in Greece on Wednesday night, Wenger was able to welcome England pair Theo Walcott and Kieran Gibbs back from injury while Ramsey played the full 90 minutes.
Source : PA
Source: PA