The Ghana international fetched the Hammers' second consolation - his third goal of the season - in the 3-2 fall to the Cherries at the Vitality Stadium.
The result leaves West Ham 11th on the table, and without a win in their last four matches.
"He [Andre] has that sense to be in the right place at the right time - That’s the reason we bought him," Bilic told his club's official website.
“He scored like that against Swansea, he scored like that against Watford and he scored a similar kind of goal again on Saturday.
"That’s a big quality for the strikers - to be in a good position, so now he’s back in good shape and of course we expect him to improve us.”
Following a club record transfer from Swansea City last summer, Ayew's West Ham career got off to a sad start, picking up a thigh injury in the league opener against Chelsea in August.
“Andre had a couple of major things which happened to him, which affected his season so far," Bilic said.
“The first was that he had a massive injury in the first game of the season, then when he came back, it was a little bit too early, so he was searching for fitness and form.
“Then he went to the Africa Cup of Nations and now he looks really good."
Saturday's goal was Ayew's second strike in three games since returning from the Afcon.