The West Ham manager was delighted with the 3-1 win over Crystal Palace, and believes the striker has matured since his first spell in England at Birmingham
Zarate was an unused substitute in the club's Premier League opener against Tottenham last weekend but Allardyce gave him his first start at Selhurst Park, and he repaid that faith with a thunderous strike from the edge of the penalty area in the 34th minute.
That set the visitors on their way to all three points, and West Ham boss Allardyce expects the 27-year-old to continue improving as he settles into life at Upton Park, following his summer move from Velez Sarsfield.
"I think the lad has grown up and matured," Allardyce told reporters. "I think he looks, at his age, a mature player with a lot of qualities in the final third, which is something different to what we have.
"He got his opportunity, he's taken it brilliantly today and hopefully he can continue to grow and improve."
Stewart Downing and Carlton Cole completed the victory, as West Ham put their disappointment against Tottenham behind them with their first win at Palace since 1991.
Allardyce added: "The quality of our finishing has been superb, that's why we're happy about the performance.
"Opening opposition up is very difficult away from home and you can see that from the quality of goals we scored to win today.
"Mauro's fantastic finish from outside the box and Stewart Downing's fantastic run and finish was of [the] highest quality that Crystal Palace couldn't defend against."