Sam Allardyce claimed Seamus Coleman's goal should have been disallowed for offside after Everton's 1-0 win at Selhurst Park left Crystal Palace in the Premier League relegation zone.
The Eagles' resilience was overcome when, with debutant Jeffrey Schlupp down suffering cramp, Coleman capitalised on the space left by his absence to collect Tom Davies' through-ball and shoot powerfully into the top-right corner.
The 87th-minute finish strengthened Everton's grip on seventh place and came after the impressive Wayne Hennessey had prevented the visitors building a commanding lead with several saves.
Replays suggest the full-back may have strayed narrowly offside when Davies played the ball, and Allardyce - who has won only one of seven games since being appointed Palace boss - feels his side were hard done by.
"He was offside, which was my suspicion when I looked at it in the dressing room," Allardyce said.
"It's two-fold: having a blip in not being able to get a sub on, playing with 10 men, and then the offside decision that could have been given by the assistant, which was an easy one.
"It's right down his side, right in front of him, but it wasn't to be. We're very disappointed we haven't got that 0-0 and clean sheet scenario we need to start building on. We'd have been delighted with a 0-0. It's difficult to take we didn't get that.
"When Jeffrey Schlupp had done a great job on Coleman, he wasn't there to do it when he scored. If Jeffrey had stayed on, I don't think Coleman would have stayed in that position."
Schlupp was on the floor and off the pitch but edged himself back on despite not being in a position to compete.
Asked if he felt referee Anthony Taylor should have allowed him to make a substitution, Allardyce said: "Yes. No doubt about it.
"But if you ask them they'll probably say it's not within the rules or something of that nature. It was a disadvantage through no fault of our own and he lets it go on a long time when he could have blown up and let us get a sub on.
"I don't think Everton have any responsibility to kick the ball out, by the way - it's up to the referee. But the big issue is the offside."
Allardyce was also asked if Schlupp contributed to Taylor's decision by returning to the pitch, and responded: "What do you expect him to do? He's lying on the floor with cramp.
"The referee would have no idea whether he did because he's right at the bottom end. The fourth official's there; knows exactly what the situation is."
The victory was Everton's third from their past four games, and came after last week's impressive 4-0 defeat of Manchester City.
Toffees boss Ronald Koeman said: "We played really well, (had) good ball possession between the lines, with good movement, and enough shots on target.
"The Crystal Palace goalkeeper was outstanding. It was difficult to stop Seamus' goal - good movement, from first to the last second he's always on the run, and it's difficult to defend that.
"Schlupp was not in that position, because he was first out of the pitch, and brought himself in the pitch. The referee took the right decision to go on, and we scored.