The hosts would have gone top with a victory but the 0-0 draw means Diego Simeone's team preserve their two-point advantage at the top, with three matches remaining.
However, city rivals Real can join them on 77 points if they can win Sunday's match against fourth-placed Sevilla.
The visitors had the best chances before half-time, with Barca – and Lionel Messi in particular – looking more threatening after the break.
Atletico's former Barca star Luis Suarez was the next to be denied by the Germany goalkeeper, before Felipe was off-target with a shot.
However, Ronald Koeman's team did most of the creating in the second half of a tight contest.
Saul was himself replaced by Joao Felix in the 68th minute, the Portugal international introduced from the bench for a third successive match as he steps up his return to full fitness following ankle trouble.
Messi came closest to finding a winner in the closing stages – but the Argentinian was narrowly wide with a 25-yard free-kick.