Palace came from 1-0 down to win 2-1 in the 93rd minute, with Michael Olise bagging late in the game after Zaha scored shortly before half-time.
It was their first away league win of the season after notching three consecutive home victories.
Speaking with the press post-match, Zaha said: “We ain’t had the best of luck with our away games. So today, me personally I let the lads know we have to play with character. We’ve got amazing talent but when we go away from home we’ve got to be as strong as we are at home and just not be bullied. Today’s result was just amazing.
“I feel like if we just play our game and stamp our authority on games we’re good enough. Obviously in some games you have to suffer but I feel like today we dominated. We gave a goal away out of the blue but we dominated and we got the points we deserved.”
West Ham took the lead through Saïd Benrahma despite Palace dominating most of the game, but Zaha pulled them level with a well-taken strike in the 40th minute.
Asked if he felt West Ham might clinch a win as the game moved towards full-time, Zaha said he always kept confidence in his team – but that a late Michail Antonio run gave him a brief scare.
“[Did you think you might concede a late goal?] Not in my head, really. There are always opportunities and, like you saw at the end, you never know. From a counter-attack goal you win the game out of the blue so there was always that thought we may get something.
“I was thinking – that [Antonio's run] came from nowhere. We had the ball then the next minute they had the ball and counter-attacked. We just needed to keep it tight but I’m glad nothing came of it, to be honest, and am happy with the win.”