Karim Benzema scored twice after the interval with a penalty and a header from a corner to give Santiago Solari's side a 3-1 lead, after Raphael Varane struck in the 34th minute to cancel out Anuar Tuhami's goal for Valladolid five minutes earlier.
The Portuguese forward then fired in the penalty in the sixth minute of stoppage time.
Cristiano Ronaldo scored twice as Real Madrid beat Athletic Bilbao 4-2 in a thrilling match at the Bernabeu to move within a point of leaders Barcelona in La Liga on Saturday.
Costa turned Juanfran's low cross into the net in the 69th minute for Atletico's third goal in the first leg of the last 16 tie against third division Lleida but immediately limped to the sidelines. He did return to the pitch, however, and completed the game despite bleeding from his knee.
La Liga's reputation as the epicenter of world soccer talent has been bruised by the world record smashing sale of Neymar to Paris St Germain and how Barcelona react to the departure of one of their leading players will define the new season in Spain.
Barcelona leapfrogged Atletico Madrid to go top of La Liga after Lionel Messi's acrobatic volley secured a hard-fought 2-1 victory against tenacious Malaga on Saturday.
Neymar made his return to the pitch after a three-month injury lay-off as Paris St Germain celebrated their eighth Ligue 1 title with a 3-1 win against Monaco at the Parc des Princes on Sunday.
Chelsea returned to their early-season form to open up a five-point lead in the Premier League, Cristiano Ronaldo and Lionel Messi continued to make mincemeat of La Liga defences and Juventus extended their lead in Serie A. In France, Olympique Lyonnais retained top spot with a 2-0 win at RC Lens.
Real Madrid coach Carlo Ancelotti will discuss his future with the club after Saturday's final La Liga game of the campaign at home to Getafe and said he wants to stay on despite a season without a major trophy.
Real Madrid have defied injuries to keep the pressure on leaders Barcelona and notched up over 100 La Liga goals but they now face arguably the toughest test of their title charge in an away match to Sevilla on Saturday.
Casemiro headed an 89th minute winner to earn Real Madrid a barely deserved 2-1 victory at Las Palmas in La Liga on Sunday as they closed the gap on second-placed Atletico Madrid to four points.
La Liga champions Barcelona began the new campaign with a 3-0 win over Alaves on Saturday with the help of a spectacular under-the-wall free kick from Lionel Messi, who scored twice, and a curling shot from substitute Philippe Coutinho.
Goals from Karim Benzema and Jese led Real Madrid to a 2-0 win over Osasuna in their King's Cup last-16 first leg at the Bernabeu on Thursday.
Valencia's Paco Alcacer, Dani Parejo and Pablo Piatti found the target as they took full advantage of the first-half dismissal of Malaga's Sergi Darder to win 3-0 at the Mestalla on Friday.
Neymar struck twice as Barca beat Atletico Madrid to reach the King's Cup semis
News of all that's transpired on and off the football field
Real Madrid kept alive their hopes of overhauling La Liga leaders Barcelona when a James Rodriguez stunner set them on their way to a serene 3-0 home victory over struggling Almeria on Wednesday.
Ballon d'Or winner Cristiano Ronaldo scored the opening goal as 10-man Real Madrid beat Osasuna 2-0 on Wednesday to qualify for the King's Cup quarter-finals 4-0 on aggregate.
Second-half headers from Diego Godin and Raul Garcia gave holders Atletico Madrid a 2-0 win over Valencia and a place in the King's Cup quarter-finals by 3-1 on aggregate.
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Lionel Messi will be working "full time" to shake off the hamstring injury sustained on Sunday that will sideline him for six to eight weeks, Barcelona's Argentine World Player of the Year said on Tuesday.
Cristiano Ronaldo hit a late double as La Liga leaders Real Madrid thumped Almeria 4-1 to extend their Spanish record for consecutive victories in all competitions to 20.
Villarreal continued their impressive revival after a season in the second division when Jeremy Perbet and Jonathan Pereira struck in a 2-1 win at home to Espanyol in La Liga on Thursday.
Now, Real Madrid have never won the Spanish treble and are fighting tooth and nail in La Liga to prevent arch-rivals Barcelona scooping Champions League and domestic league and Cup titles for a second time.
News of all that's transpired on and off the football field
Real Madrid included Denis Cheryshev in their starting lineup for their first match of the 2015-16 competition even though he should have been serving a one-match suspension and the Russia forward opened the scoring in a 3-1 victory.
Real Madrid top scorer Cristiano Ronaldo has signalled strong support for coach Carlo Ancelotti and sent a clear message to club president Florentino Perez that he does not want the Italian to be sacked.
Barcelona made a brisk start to the season with new coach Luis Enrique at the helm in place of Gerardo Martino and a host of additions to the squad, including midfielder Ivan Rakitic and goalkeeper Claudio Bravo.
Atletico Madrid kept pace with leaders Barcelona and restored their five-point lead over third-placed city rivals Real when Diego Costa and David Villa each scored a penalty in a 2-1 win at Granada in La Liga on Thursday.
Raheem Sterling struck twice as the Premier League champions Manchester City beat Crystal Palace 3-1 at Selhurst Park on Sunday to stay within two points of leaders Liverpool.
News of all that's transpired on and off the football field
Barcelona seized control of their King's Cup quarter-final tie against Atletico Madrid when Lionel Messi followed up his saved penalty to score and secure a 1-0 first-leg victory at the Nou Camp on Wednesday.
This and more from the happenings in the world of football
News of all that's transpired on and off the football field
News of all that's transpired on and off the football field
Barcelona's Lionel Messi was at the heart of another devastating attacking show.
Fernando Torres stole the show with a dramatic double as Atletico Madrid secured a battling 2-2 draw at Real Madrid on Thursday that dumped the holders out of the King's Cup 4-2 on aggregate.
Lionel Messi dragged Barcelona back from two goals down by scoring twice to snatch a 2-2 draw at home to Valencia in a thrilling encounter.
Manchester City swept aside second-tier Cardiff City 2-0 with first-half goals from Kevin De Bruyne and Raheem Sterling on Sunday to reach the FA Cup fifth round in their quest to win an unprecedented four trophies this season.
A summary of the results of the major European football league matches on the weekend.
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