Substitute Calvin Andrew struck a second-half winner at Hillsborough on Saturday after Wednesday drew level through a Clint Hill own goal just before the break following Neil Danns' opener.
"The lads have been brilliant this week, in the circumstances," said Warnock. "It is difficult for them, when they're not getting paid, but their spirit is very good.
"Me and the players haven't been paid but we will get paid. It's not easy for the chairman because he's put a lot of money in over the years
"We've just got to get on with it because there are a lot of worse things happening than just not getting paid for a few days.
"I thought we deserved to win and it's just great to be in the hat for the next round."
Danns gave Palace a 19th minute lead, finding the net home from close-range after the Wednesday defence failed to clear following a shot from Sean Scannell.
But luck was on Wednesday's side when they equalised just before half-time. Leon Clarke's angled shot was going beyond the far post but Palace defender Hill turned the ball into his own net.
Andrew scored the decisive goal, his low shot going in off a post after Frank Simek failed to cut out a long ball.
Speroni was also tested by a long-range strike from Marcus Tudgay during a good spell for the home side.
Owls caretaker manager Sean McAuley was disappointed with his side's display after a result which takes their run to 12 games without a win.
"There were certain things in the game we weren't happy with," said McAuley. "I thought they got the goal against the balance of play, but they had more commitment and desire."
McAuley is under no illusions about the size of the task facing him or whoever takes charge on a permanent basis at Hillsborough.
He said: "Whoever takes this job needs to work incredibly hard because we need play-off form between now and the end of the season.
"I'll be definitely be looking to strengthen the squad. Just because it's a temporary position doesn't mean to say that work isn't being done."