The Catalans will play Valencia or Real Betis in the final on 25 May.
Questions have been raised over Suarez's form and future since the Christmas break. He had been goalless in five games until the 4-2 win over Sevilla last Saturday.
In that La Liga match, skipper Lionel Messi was the star with a hat trick and the assist for Suarez's strike.
However, at the Santiago Bernabeu on Wednesday night, it was the 32-year-old Uruguayan who muscled into prominence.
"When he is in the area he is at 200 per cent," said Barcelona coach Ernesto Valverde said. "He has a sense for where the goal is."
The victory over their arch rivals keeps Barcelona on course for three trophies. They are seven points ahead of Atletico Madrid with 13 games remaining in La Liga and also in the last 16 of the Uefa Champions League.
"We want to win everything," Valverde said. "That is what our fans want."
Real Madrid, who are also in the last 16 of the Champions League, need Barcelona and Atletico Madrid to slump to have a chance of the domestic championship.
But they can make inroads into Barcelona's lead on Saturday when they entertain the pacesetters. Revenge will be the aim after the annihilation at the Camp Nou which effectively terminated the humiliatingly short reign of Julen Lopetegui.
"We played well," said Madrid midfielder Casemiro after the Copa del Rey semi-final. "The result was not good but we will be waiting for them on Saturday. The result was not a fair reflection of the game."
Vinicius Junior appeared to be Madrid's most likely source of a breakthrough. But the 18-year-old missed an array of chances before Suarez claimed the opener just after the pause.
Varane's blunder made it 2-0 to Barcelona and 3-1 on aggregate. With the away goals rule in operation, it meant Madrid would have to win outright.
All hopes of a comeback ended 17 minutes from time. Suarez, tripped by Casemiro in the penalty area, picked himself up and converted the spot kick to seal the victory.
"We are sad because we wanted to play in the final but we go out with our heads held high," said Madrid coach Santiago Solari. "We did everything we could. We didn't take our chances and they took theirs."