The Hammers needed an injury-time equaliser to seal a 2-2 draw with the Cottagers as Junior Stanislas' deflected effort earned Zola's side a point at Upton Park.
Fulham debutant Kagisho Dikgacoi saw red in the first-half for violent conduct but the hosts failed to capitalise despite being a goal up at the time of the dismissal.
The Cottagers raced into a shock lead after coming from behind soon after the break as the Hammers looked more like the team a man short.
Stanislas' late strike salvaged a point for West Ham as resilient Fulham's defence finally broke down, but Zola admits he was not expecting to struggle after the first-half dismissal.
"After the first-half I was expecting different things in the second," Zola told Sky Sports.
"The first-half was really encouraging and the way the team played the ball was very good.
"The start of the second-half was not good enough and this is not the first time that's happened.
"We were not switched on because we cannot concede goals in a game that in my opinion was already closed.
Mistakes"Their second goal was again through individual mistakes and that cost us very much.
"I don't believe a game can be won until the end. Arsenal today were losing twice and they ended up winning 6-2 - that's the kind of belief we need to have."
The result halted a four-game run of defeats for West Ham but with their inability to take three points after playing over half the game with an extra man, Zola admits he has cause for concern.
"I'm concerned as much as I have to be. I know that even if we're playing good football, as we are right now, it's not enough.
"We have to make sure we convert that into points. We are half-way there but we need to improve a few things to make sure that the points come in."
The former Chelsea player has ruled out spending in January, insisting that his current squad is good enough for him to work with.
"I'm happy with the squad I've got. I decide on them and I will work with them until the end and see what happens."
Teenager Stanislas, who is a product of the highly acclaimed West Ham youth academy, came off the bench to score the equaliser and Zola highlighted his positive approach to the game.
Zola added: "I'm pleased for him because he came on with the right attitude and made an impact on the game."