Palace have won five of their last six matches to build a solid cushion above the bottom three, having also beaten league leaders Chelsea at Stamford Bridge in a strong run of form under Allardyce.
Andros Townsend opened the scoring for the hosts, with Yohan Cabaye strike adding a sublime second goal and a Luka Milivojevic penalty wrapping up what Allardyce felt was a fully-deserved three points for the Eagles.
"I think that we had the opportunities to score goals and tonight tactically the players were aware of what had to happen to beat Arsenal," a delighted Allardyce told Sky Sports.
"The first thing was to defend correctly as an 11 and frustrate them and use the ball into the spaces and into [Christian] Benteke to fill the area and win the knock downs and get balls into the box. Defensively they leave [Shkodran] Mustafi and Gabriel exposed and they're left on their own so we can get in their box and get chances.
"Today with the work rate and determination to log on to their possession and create chances, it was down to us shutting down throw-ins, which people don't see but that was the basis of us getting so many attacks, and then quality finishing.
"Our performance was not a shock as we won at Chelsea, but we knew if we repeated that tonight we could get a result and we did that.
"We all know [Arsenal] are in a poor spell of results for the first time in years and years so every player, top or not, feels the frailties when form and results are not good. But the only way to take advantage is to work hard and shut them down to frustrate them.
"We didn't let them play and everything worked perfectly, the lads carried out the game plan perfectly and to boot we got a clean sheet against an Arsenal side that scores a lot of goals."
Despite Palace sitting six points above the drop zone after a third successive home victory without conceding a goal, Allardyce does not believes his team are safe and is targeting another victory in their next game against Leicester City.
"We've got Leicester next week and we look for another three points to draw away from the bottom three," Allardyce added. "We had four wins on the trot, then a loss [to Southampton] and another win, and it shows how hard it is to get away. We're still not clear having won five out of six but if we keep up this level of performances we'll be okay.
"We probably need another three points but as each game comes we have to try and keep the points up. Most important thing was we had games in hand but had to get the points and we got more than people expected today, people said a draw would be good, but we got a deserved three points.
"Two of the hardest teams are out of the way - Chelsea and Arsenal - and we beat them both. We're in a very good mood at the moment and I hope the lads can keep it up. Today it was done with a few players injured and they're coming back, everyone can enjoy tonight and get back to work tomorrow morning."