Zaniolo struck to cancel out Michail Antonio's opener for the Hammers.
He later said: "It was a very good match.
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"The first half they were more aggressive but we played well in the second half. Our team is very good. We competed well today and we have to fight.
"It will be very hard (the top-four fight) but we are a very good team. We have a very good coach and we prepare very well."