Villarreal beat Huesca 2-0 to go through 4-3 on aggregate, Celta Vigo downed Almeria 1-0 to win 4-1 over two legs and Granada won 1-0 against Leganes to progress on away goals having drawn the tie 2-2.
Partey volleyed into the bottom corner from 25 metres after 53 minutes as a second-string Atletico dominated against their lower league opponents but lacked the finishing touch in the first half.
The 2013 King’s Cup winners are in fine form having won nine straight matches in all competitions. They are joint top of La Liga with Barcelona on 35 points and have reached the knockout stage of the Champions League.
“We knew that it wouldn’t be an easy game and with the stadium half empty we had to be motivated," Atletico keeper Miguel Angel Moya told reporters.
“We had chances which we didn’t take but kept going and the goal came.”
Atletico coach Diego Simeone made nine changes from the side that beat Athletic Bilbao on Sunday but the team quickly found their rhythm and pushed Reus back.
Fernando Torres, a goal short of his 100th in an Atletico shirt, had three clear chances but was denied by excellent saves from Edgar Badia while a Koke freekick from the left wing came back off the right post after 32 minutes.
Defender Lucas Hernandez appeared unlucky to have a goal ruled out for a supposed foul at a corner but after Thomas put them ahead, Atletico eased off for a comfortable win.
Villarreal substitute Roberto Soldado struck after earlier missing a penalty to put the seal on a 2-0 victory over Huesca.
Having lost the first leg 3-2 away, Villarreal pushed forward from the start and a deflected shot from Manuel Trigueros put them ahead on 28 minutes.
Soldado had been rested from the start but came on and hit the crossbar with a penalty after a foul by David Morillas on Matias Nahuel.
He made up for it though by slotting home from a Trigueros pass after 78 minutes.
A Daniel Wass freekick gave Celta Vigo a 1-0 victory over Almeria and a Youssef El-Arabi goal gave Granada a 1-0 win over Leganes.
Barcelona are already in the last 16 having beaten Villanovense at the start of the month after the tie was brought forward due to their participation in the Club World Cup.
Real Madrid were thrown out of the competition having fielded an ineligible player in their first-leg game against Cadiz.
(Reporting by Tim Hanlon in Barcelona, editing by Toby Davis)
