But they also had England goalkeeper Robert Green to thank for the win after he saved a 49th-minute penalty from Ashley Young.
Young atoned for his blunder with an equaliser two minutes later but Zola was delighted with a win that takes them up to 16th position.
"The team is playing better and this may well be the turning point," said Zola.
"We demonstrated desire and belief and that's what we wanted. Everything else is there. It was a fantastic game to win tonight.
"We got the reward we wanted in the end. I believed and so did the players.
"Every time I play Hines he's been very good. He just needs to keep working the way he is doing. I am very pleased for him and for us."
West Ham were hampered by the loss of Herita Ilunga and England striker Carlton Cole in the first half. Both players collected hamstring injuries and that could rule Cole out of England's forthcoming friendly with Brazil in Qatar.
"It was unbelievable," said Zola. "The way we started the game was perfect and then we lost Herita and Carlton and the atmosphere went a little bit flat among the players.
"But the second half was a different story and we got our reward.
"I don't think the injuries are serious but they are both hamstrings. We will assess them in the next couple of days."
Zola expressed his astonishment at Villa's penalty award which Steve Bennett awarded for Manuel Da Costa colliding with James Collins after he had headed the ball clear.
Luckily for Zola, goalkeeper Green made a superb save from Young.
"I saw only one penalty," said Zola.
"The second one I could not understand. Da Costa jumped high and found James on the way down.
"It was a little bit too much. Green has made a couple of great saves tonight and I was delighted for him.
"It is a long way to go but the team is alive and wants to be out of the situation we are in. We are ready to fight and we will do that. Tonight is just one step."
Villa boss Martin O'Neill was disappointed in the manner that his side lost the game and would not give his view on Habib Beye's late red card for a second bookable offence or the penalty award to West Ham.
"We got a bit tired towards the end of the game and we got beaten in the last seconds," said O'Neill. "I am very disappointed.
"I don't want to talk about the decisions.
"It is a pointless exercise. Anybody who has access to a TV monitor can make their own minds up. I have not spoken to the referee.
"I am disappointed to have lost. I think we did enough to get something out of the game. We were in the ascendancy in the first half and their goalkeeper made some good saves.
"We looked a very good threat and thought we would go on to win it. You can lose a goal at any stage but it is disappointing to lose at that stage. We looked tired in the last 15 minutes but it is not an excuse."
Villa now lose Stiliyan Petrov for the next game after he picked up his fifth booking and that also upset O'Neill.
"He is suspended now," added O'Neill.
"He was magnificent last season and has kept that form again. He is a really fine player."
West Ham 11/8, Everton 15/8, Draw 12/5