Lamine Yamal scored before half-time and Ronald Araujo added a second goal early in the second half to put Barcelona in control of the quarter-final on Tuesday.
The 32-time cup winners were taken all the way by their second division opponents, who had beaten Real Madrid in the previous round.
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Albacete, scored late through Javi Moreno but could not force extra time.
"The last few minutes... it's normal, they got a goal, the cup is like this," Araujo told Barca's media channel.
"I'm very happy (to start again), it was an important step... a hug to the coach, he helped me a lot."
Barcelona controlled possession in the first half but had difficulty creating clear chances.
"It was a tough match at the end, it was very hard. We could have made it much easier for ourselves, we missed many chances, but I think also we deserved (to win)," said Flick.
"I'm happy, proud of the team and we're in the semi-finals, and that's the best thing."
Rashford won the ball back, and Frenkie de Jong’s pass reached Yamal, who scored.
Albacete threatened before the break when Antonio Puertas broke through, but goalkeeper Joan Garcia collected the ball.
The goalkeeper accidentally conceded a corner when he walked off the pitch holding the ball to pat Araujo on the head.
After half-time, Dani Olmo missed a chance from close range after a pass from Rashford.
Barcelona doubled their lead when Araujo headed in a corner from Rashford. It was Araujo’s first start since returning from a month away from the team.
Albacete increased pressure in the final stages.
A goal by Puertas was ruled out for offside, but Moreno scored in the 87th minute with a header from a Jose Lazo's free-kick.
Barcelona defended through stoppage time, with Gerard Martin clearing a shot off the line.
Barcelona coach Hansi Flick said his team missed chances but deserved the win. Araujo said he was happy to return and help the team.
"We have to finish with more conviction," said Flick, who said he was happy that Araujo netted on his return to the starting line-up.
"He scored a very important goal for us, it could help him to get more confidence and believe in himself," added the German coach.
In the other quarter-finals Valencia welcome Athletic Bilbao and Real Sociedad visit Alaves on Wednesday, while Atletico Madrid travel to face Real Betis on Thursday.