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After a poor run of form left them on the fringes of the Premier League relegation zone, Palace played with renewed confidence to deservedly beat a lacklustre Watford side.
They now have the chance to avenge their defeat by United 26 years ago, when they lost a replay 1-0 after a thrilling 3-3 draw in the first game.
Palace manager Alan Pardew, whose goal helped Palace beat Liverpool in the 1990 semi-finals, said his team had been worthy winners of a scrappy match.
"Watford had five or 10 minutes when they bossed the game and scored, but we bossed it for 80 minutes and that is a testament to our players mentality," Pardew, who took as manager midway through last season, said.
"We will have a different game plan against Manchester United, but that is not to say we can't control the game. We deserve to be there."
Watford manager Quique Flores, who has found himself under pressure despite ensuring Premier League safety, said his side were unable to get on the front foot after Bolasie had given Palace a sixth-minute lead.
"It was hard to concede the first goal," he said. "It took some time to recover. We came back and in the second half we matched Palace.
"We have a lot of reasons to be happy this season. The main target was to stay in the Premier League."
The powerful Bolasie and Wickham have both been absent for lengthy spells with injuries this season and they showed Palace fans what they had been missing.
Bolasie headed in Yohan Cabaye's corner to put Palace in front after six minutes and they should have built on their advantage before halftime.
Watford suffered a blow when midfielder Etienne Capoue sustained a knee injury and had to be replaced by Mario Suarez.
Troy Deeney sparked a tense semi-final into life when he headed Watford's equaliser nine minutes after halftime but they were only level for seven minutes.
Wickham stretched to meet a cross from Pape Souare and his header flew beyond the reach of flat-footed Watford keeper Costel Pantilimon, who was slow to react.
Watford mounted late pressure and there were a few anxious moments for Palace in stoppage time but they deservedly held on to advance to their second Cup final.
Manchester United beat Everton 2-1 on Saturday.