Gareth McAuley scored in injury time to snatch a point for the away side at the London Stadium.
Ayew resumed club training last week after helping Ghana reach the semi-finals of the 2017 Africa Cup of Nations, and had been expected to play a part on Saturday.
“Andre Ayew came back on Friday,” Bilic said.
"He had a long Africa Cup of Nations – they didn’t play more games than we played here so it’s not physical, but psychologically, it means a lot to them and it drained him.
“They were there until the end – they played for third place – so we gave him a few days off and he came back on Friday.
"That’s why he was not in the squad on Saturday.”
Ayew joined West Ham at the start of the season, but could not get off to a good start following a thigh injury on his league debut, effectively sidelining him for two months.
He returned in late October but his struggle for form has affected his chances of securing a permanent starting place.
The 27-year-old made a dozen appearances, involving six starts, ahead of his departure for the Afcon.
The French-born is expected to return to league action when West Ham play as guests to Watford on February 25.