Roberto Soldado's 53rd minute goal gave Getafe a 1-0 victory, but Sevilla brought a 2-0 advantage into the game from the first leg, and held on in the second half .
They will now likely face Atletico Madrid in the final, as Los Colchoneros head into their second leg against Racing Santander with a commanding 4-0 advantage from the first meeting.
Diego Forlan scored twice from the penalty spot for Atletico.
The Rojiblancos were aiming to silence their critics tonight, having been booed off by their fans at the weekend in a 2-0 Primera Division loss to Malaga.
And it took just nine minutes for Simao to fire Atletico in front with a first-time strike into the roof of the net.
Jose Antonio Reyes then doubled the lead five minutes before the interval, tapping home from close range after the Racing defence had failed to clear the ball.
Forlan scored the first of his penalties after 62 minutes, although replays suggested Toni Moral's challenge on Jose Manuel Jurado took place outside of the area.
But there was no doubt over the second spot-kick eight minutes later, with Forlan again providing the finish after Sergio Aguero was sent to ground by Marc Torrejon.
Aguero should have added a fifth three minutes into stoppage time but he somehow turned Tomas Ujfalusi's inviting cross past the left post from a few yards out.
Sevilla hold a 2-0 first-leg advantage over Getafe in the other semi-final.
The hosts were unfortunate to find themselves a goal down midway through as Celta had had at least as much of the play.
Forlan intercepted a sloppy ball out of defence by Francisco Noguerol to get half a metre ahead of his marker and slotted home clinically to give Atletico a 2-1 win on aggregate.
Getafe made it through to the semi-finals despite going down 1-0 to Real Mallorca on the night.
Getafe had been in pole position to progress after claiming a 2-1 win in Mallorca last week.
Aritz Aduriz had a chance to put the visitors ahead midway through the first half but his spot-kick was saved.
However, the 28-year-old made up for his error when he headed home in injury time at the end of the first half.
They could not find a further goals after half-time and Getafe went through on away goals.
Racing triumphed 3-0 on the night after going in with a vulnerable 2-1 home win in the first match.
Xisco eased the pressure early on with a sixth-minute effort and defender Henrique made things comfortable when he added a second after 22 minutes.
The hosts could not find a way back into the fixture and it was instead left to Sergio Canales to add a third after 78 minutes.
Sevilla lost 1-0 at home to a battling Deportivo La Coruna, but will take their place in the last four having eased to a 3-0 win in the first meeting.
Seville-born striker Rodolfo Bodipo was the scorer for Depor but the visitors could not add to his 43rd-minute strike in the second half.
Roberto Trashorras struck early on to give the struggling Segunda Division side a shock lead.
Tiago levelled for the hosts eight minutes later, but the Primera Division side could not find a winner.
In today's other first leg, Racing Santander beat Osasuna 2-1 at El Sardinero - but it could have been better but for a late goal from the visitors.
Goals from Gonzalo Colsa on the hour mark and substitute Papa Kouli Diop six minutes from time appeared to have put Racing in control, but Walter Pandiani's goal in the 88th minute set things up nicely for the second leg.
Sevilla, the 2007 winners who last week dumped out holders Barcelona at the last-16 stage, opened the scoring in the 26th minute when Alvaro Negredo headed home Spain team-mate Jesus Navas' cross.
Renato Dirnei then doubled the Andalusian side's lead in the 67th minute following good work by substitute Diego Perotti, and two minutes later Navas added a third to make Sevilla overwhelming favourites to progress in next week's home leg.
Manolo Jimenez's side could have recorded an even bigger win but for some indifferent finishing and the heroics of home goalkeeper Manu, while the closest Depor came to scoring was when Juan Rodriguez struck the woodwork in the first half.
Sevilla also knocked Depor out of the cup last season after recording a 5-1 aggregate win over the Galician outfit at the last-16 stage.
In the evening's earlier match Real Mallorca saw their perfect home record come to an end as they were beaten 2-1 by Getafe.
Mallorca went into the match having won all nine of their Primera Division matches and both of their Copa del Rey ties at their Ono Estadi ground this season, but they came unstuck against a Getafe side who are looking to reach the final for the third time in four years.
All three goals came in the second half, with Manu del Moral and new signing Miku - making his debut after joining the club last week having been released by Valencia - putting Getafe 2-0 up by the 66th minute.
Mallorca gave themselves a boost for the second leg though when Gonzalo Castro pulled a goal back in injury time.
The remaining two quarter-final first legs take place tomorrow, when Atletico Madrid host Celta Vigo - the only non-Primera Division side remaining in the competition - and Osasuna go to Racing Santander.
The second legs of all four quarter-finals are scheduled to be played next week.
Trailing 3-0 from the first leg, Atletico quickly got to work as Simao opened the scoring in the 21st minute before Sergio Aguero added a second three minutes later.
Tomas Ujfalusi added a third in the 39th minute and Aguero's second just after the hour mark had Atletico in front.
Carlos Carmona put the tie back in doubt with a 71st-minute reply and things got worse for Atletico when Assuncao got his second yellow card two minutes later.
But Simao's 84th-minute strike sent the Madrid club through.
Real Mallorca advanced thanks to a dramatic last-minute goal from Mario Suarez as they also overcame a first-leg deficit against Rayo Vallecano.
Trailing 2-1 from the first leg, Mallorca's task got all the harder when Piti put the visitors in front on the stroke of half-time.
But Victor hit a brace in the 50th and 60th minutes to level the scores on aggregate before Suarez struck.
Rayo Vallecano beat 10-man Mallorca 2-1 with goals from Michel Sanchez and Juan Collantes coming either side of Julio Alvarez's strike for the visitors.
Mallorca had Felipe Mattioni (pictured left) sent off in the 36th minute with the score at 1-1.
Osasuna did not have the excuse of being a man down as they also lost 2-1, going down away from home to Hercules.
Joseba Del Olmo got the winner in the 68th minute after Jorge Galan had cancelled out Hidalgo Cristian's opener.
In Thursday's only all-La Liga clash, Malaga took a 2-0 lead over Getafe thanks to first-half strikes from Antonio Apono and Arnaldo Edinho, but handed some of the initiative back ahead of the second leg when Milan Stepanov scored an own goal midway through the second half.
