Lucas was replaced at half-time after suffering what Atletico described as a "strong facial trauma" in a 1-0 defeat away to Sporting CP, Diego Simeone's men heading through 2-1 on aggregate.
Atletico later confirmed the full-back had been taken to hospital for tests, which ruled out any fracture.
Costa followed shortly after the restart after the striker appeared to sustain an injury to his left leg, but 2010 and 2012 winners Atletico were able to cling on.
The club later confirmed on their website that Costa had suffered an injury to the back of the thigh and that he would undergo tests to determine the severity when the team return to Madrid.
Simeone was pleased with his team's response in the match, especially after losing the duo to injury.
"We're very happy because once again, the team reinvented itself," Simeone said after the match. "Lucas and Costa got injured and the team kept on playing with a massive amount of effort."
Injuries to Costa and Lucas come as a blow to Simeone, who saw his squad options reduced by the sales of Nicolas Gaitan, Augusto Fernandez and Yannick Carrasco - who all moved to the Chinese Super League - during the January transfer window.