Ghana striker Ayew left his struggling side in the lurch when he was dismissed for a vicious elbow into the face of Aaron Cresswell after just 17 minutes.
Rock-bottom Villa had been on top at that point, having forced a couple of corners and seen a reasonable penalty shout turned down when Gabriel Agbonlahor's cross hit Michail Antonio's arm.
So it was utterly mystifying when Ayew, who was jostling for position with Cresswell from a throw-in deep in West Ham territory, swung out his elbow into the defender's face and left him sprawled on the floor.
And it was Villa who ended up with a bloody nose after Antonio and Cheikhou Kouyate struck in the second half to end the 10 men's resistance and shove them closer to relegation.
Garde refused to discuss what he said to Ayew afterwards, and also declined to answer questions about his position, the lack of January transfer activity which has angered fans, and the away support chanting against the club's board.
"It will remain private, between him and me, what I said to Jordan. To speak shortly after the game is not always the best solution," Garde said.
"I'm not going to speak any more (about) the incident, it's a major one in the game but I don't have to explain my feelings, and you can guess Jordan's feelings. Whatever happens in the dressing room is private.
"But I'm not sure it was the only incident that turned the game. Of course I think we should have had a penalty.
"I'm not a lucky or unlucky man, but I don't like to hear it's because we are at the bottom. The rules in football are there for everybody."
Patience was the order of the day for West Ham and they finally got their reward just before the hour mark.
Mark Noble floated a long, diagonal ball into the area, Antonio got in behind the Villa defence and guided a deft header back across Mark Bunn and into the far corner of the net.
The killer second came with five minutes when Enner Valencia raced clear before teeing up Kouyate.
While Villa look doomed, the Hammers are still looking up, they are still just a point behind fifth-placed Manchester United.
"It's a massive win for us," said manager Slaven Bilic. "I have nothing but praise for my players, and I have to congratulate Villa's players for fighting and not giving up.
"Until the red card they were confident. Unfortunately for them that moment of madness affected the game a lot.
"In the second half we played exactly the way teams should play. We didn't panic, we were patient, dynamic and quick, but without losing composure and shape.
"We played really good. We are becoming a serious team."