Pique, whose Spain international team mate and goalkeeper Victor Valdes was ruled out for seven months this week after he tore knee ligaments, had to be replaced by Marc Bartra after only 12 minutes of the quarter-final first leg at the Nou Camp when he landed awkwardly after jumping for the ball.
Barca said Pique had damaged his right hip and would be out for around four weeks, meaning he will miss next Wednesday's second leg in Madrid, the Copa del Rey final against Real Madrid on April 16 and several La Liga games.
He may also have to sit out the two-legged Champions League semi-final at the end of the month if Barca qualify.
Costa, who had scored seven goals in five appearances in this season's Champions League before Tuesday, and has 25 in 30 La Liga outings, pulled up clutching the back of his right leg with half an hour gone and was replaced by Diego.
The game ended 1-1 after goals from Atletico's Diego and Barca's Neymar.
Atletico coach Diego Simeone told a news conference Costa would need more tests to determine the extent of the injury and how long he would be out of action.
Raul Garcia, David Villa or Adrian were ready to stand in for the Spain international in the meantime, Simeone added.
Simeone's Barca counterpart and Argentine compatriot Gerardo Martino was pleased with the way Bartra played after coming off the bench.
The 23-year-old Barca academy graduate, third in the pecking order behind Pique and Javier Mascherano, turned in an accomplished performance, although his task was made considerably easier when the combative Costa was forced off.
"In these situations the best thing is to be able to count on all the players in the squad," Martino told a news conference. "Bartra did very well, he was very precise.
"I was not surprised. He is a centre back with a lot of qualities. What I was surprised by was how quickly he adapted to the game almost without having warmed up.
"He did it with a lot of personality."
Both Martino and Simeone appeared content with the draw after an entertaining tussle which Barca increasingly dominated as the second half wore on.
Diego put Atletico ahead in the 56th minute with a long-range stunner that surprised Barca goalkeeper Jose Manuel Pinto before Neymar levelled in the 71st when he galloped onto a brilliant Andres Iniesta through ball and curled a shot into the far corner.
It was the fourth stalemate between the Spanish rivals this term after they drew 1-1 and 0-0 in the two-legged, season-opening Spanish Super Cup and 0-0 in La Liga in January.
"I am full of pride at the way in which we played and if we continue like this there are chances for success," Martino said.
"After (Atletico's) goal you saw the best of Barcelona. In the second half we moved the ball about well and played with depth.
"Our intention in every game is to go out and win, so we will have to go (to Madrid) and do what we always try to do."
Simeone added: "We have a team with huge heart and it made me so happy to be able to compete against these opponents.
"From a tactical point of view, it was a beautiful game. Both teams played to their strengths and stayed true to their style."
Atletico, who last won La Liga in 1996 with a team featuring Simeone, are a point clear of Barca at the top with seven games left and their final match of the campaign next month is another trip to the Nou Camp.
It is the first time any club has mounted a genuine challenge to the domestic dominance of Barca and Real Madrid, the world's richest clubs by income, since Valencia won in 2004.
- Sports & Recreation
- La Liga
- Diego Simeone
- Real Madrid
- Marc Bartra
- Gerardo Martino