Schneiderlin scored his first goal for his new club, coming 511 days after he scored his last Premier League goal, ironically against Everton.
Everton continued their chase of the Premier League's top six with a comfortable 3-0 win over West Brom.
The result sees Everton stay seventh but close the gap on six-placed Manchester United to two points, although Jose Mourinho's side have two games in hand, while the Baggies remain in eighth place.
A Lukaku deflected pass rebounded into the path of Ross Barkley at the edge of the box, and his drilled shot was spilled by Ben Foster, with Mirallas acting quickest to smash into the roof of the net.
It was at the mid-point of that bout that the advantage turned in Bellew's favour when Haye picked up an injury, though there was little chance of a similarly-timed momentum swing for the Baggies despite Baines' withdrawal ahead of the second half.
That situation has been frustrating for Everton who, despite just one defeat in the league this year, haven't managed to move any higher than seventh place.
Barkley assisted Romelu Lukaku's third goal from a "free role" and that capped off the victory after first-half strikes from Kevin Mirallas and Morgan Schneiderlin.
It also showed that the Blues would only be seven goals worse off in the goal difference column without Romelu Lukaku's 18 league goals, highlighting this side may not be as dependant on the Belgian as first feared. Subs: Funes Mori 5, Gueye 5.
Whilst Rondon is renowned for making unselfish runs into the channels, the Napoli-owned hitman is an out-and-out poacher, and would surely benefit from the space created by the aforementioned West Brom forward. Lukaku was played in and he whipped a right-foot shot just wide of the far post. They weren't bad in that match, but conceded two avoidable goals.
West Brom were finally finished off eight minutes from time - Lukaku rising to meet Barkley's cross and heading beyond the helpless Foster.