Palace's 3-2 win at home to arch-rivals Brighton last weekend lifted them six points clear of the relegation zone, with just under a month of the season remaining.
Zaha bagged two of the goals in the first 24 minutes and is continuing to be a massive influence for Roy Hodgson's men since recovering from injury.
And the Ivorian winger is convinced he is improving as a player at Palace and sees his long-term future in south London.
"I have felt that with the performances we have been producing that eventually the results were going to come," he said. "I have had faith in the team but obviously the injuries didn't help, but now we are showing what we are capable of.
"I put the pressure on myself because I want to be that player anyway, when you look at different teams they look to someone to score the goals or do something special and personally I like to have that on me, and it makes feel a better player and I want to prove that I am that kind of player.
"Crystal Palace is my home town and the support we have here is genuine support, with families watching with their kids loving us as players so I don't really see myself anywhere else.
"I am enjoying my football here and just focussed on us being a stable club and not having to listen to all the speculation about us getting relegated."