Messi chipped in a penalty to reach another astonishing milestone at Camp Nou but twice Atletico hit back with spot-kicks of their own, Saul Niguez converting both after Barca took an early lead through Diego Costa's own goal.
Atletico stay third, two points ahead of Sevilla, who won 3-0 at Leganés.
With a superior head-to-head, it would take a dramatic capitulation for Real Madrid not to lift the trophy from here but there is no guarantee Barcelona can finish strongly enough even to create pressure.
This was their third draw in four matches and with suggestions of disharmony between the players and coaching staff, the future of coach Quique Setién looks bleak too.
"From the dressing room I feel supported, yes," he said. "And regarding the meeting with the board, I will not say what was said in that."