Liverpool fan Coady steered a header into the net from Ruben Neves' excellent cross on 49 minutes to give the visitors the lead, and Everton finished the game with 10 men when Jonjoe Kenny received two yellow cards.
Everton are in 17th place on 22 points from 26 games, only out of the relegation zone on goal difference, though they do have games in hand on the teams around them. Wolves moved up to seventh with 46 points from 29 matches.
"It was massive. We knew there was going to be a reaction [from Everton]," Coady told reporters. "No matter what you say about them, they are a top team and a top football club.
"It was important we came here with the right mentality and know that when the crowd gets up it's tough to play against. We had to frustrate them at times and I thought we did it and got the goal in the second half which is fantastic."
Everton are next in action on Thursday when they host Newcastle United, desperate to find a way to win as the spectre of a first relegation since 1951 looms larger after only two wins in their last 20 Premier League games.
They started brightly, roared on by a desperate home crowd, and might have gone in front as Richarlison got in behind the Wolves defence, but his shot from a tight angle was saved by goalkeeper Jose Sa.
Demarai Gray had a close-range effort saved too, while the visitors grew into the game as the half wore on.
Richarlison hit the side-netting with a snap-shot, which was as close as the home side came to finding an equaliser.
"There's no hiding place. We've got to dig deep," Everton defender Mason Holgate told Sky Sports. "I thought we showed spirit and fight but ultimately when you are down there you have to pick up points.
"We're disappointed but we have to start winning games. If we don't start winning games we have to start not losing games."