Sunderland moved out of the relegation zone with the point gained from the goalless draw against Arsene Wenger's side but Allardyce believes the performance from his side deserved more.
Sunderland hit the woodwork through Patrick van Aanholt's first-half free-kick, while Jermain Defoe was denied a penalty after Per Mertesacker handled in the area and Petr Cech made a smart double save to deny Defoe and Yann M'Vila.
Arsenal had their chances too, though, with Alex Iwobi also denied a spot-kick after DeAndre Yedlin appeared to handball in the area, while Vito Mannone made some important stops in the Sunderland goal.
But Allardyce felt his side edged the game on the way to their third clean sheet in four Premier League games.
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"Slightly disappointed we haven't taken all three points," admitted Allardyce when asked whether he saw it as a point gained or two lost.
"On the overall performance, if you had said before the game or at half-time would you be happy with a point? Then yes.
"But at the end of the game and on the back of a slight change in tactics at half-time and a much better second half, I was slightly disappointed we didn't get the one goal to pinch it.
"Another clean sheet for us though and a very good point.
"For the first time in many months it moves us out of the bottom three on goal difference and psychologically hopefully it is a big boost for our players as it is in our own hands.
"Can we maintain that level of performance and produce it at Stoke City next week?
"For me, another clean sheet is where our safety really lies. If we can keep as many clean sheets as we can because we have done really well on that.
"It has helped us claw our way out of the bottom three recently and we need the same next week.
"I am really pleased with the overall performance of all the players.
"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.
"I was hoping we would score in that period but we couldn't manage it," added Allardyce.
Sunderland travel to the Britannia Stadium to face Mark Hughes' Stoke City side next Saturday.