Antoine Griezmann missed a good chance to give Barca the lead before halftime but hit the side netting while midfielder Frenkie de Jong should have scored late in the second half but missed the target from inside the area.
The teams went to the dressing room tied 1-1 at the break as neither could find a second goal for the remainder of the first half.
The struggle to find a go-ahead goal raged through the second half as Ronald Koeman brought on new arrivals Miralem Pjanic and Dest, who was making his debut for Barcelona after completing his move from Ajax ahead of the weekend.
Dest, 19, becomes the first American player to play for Barcelona’s senior team.
Sevilla could also have snatched a winner, with Youssef En-Nesyri seeing a deflected shot hit the crossbar before failing to test Neto with a free header.
Koeman’s changes wound up being in vain, though, as neither team found a winner with Barcelona and Sevilla remaining unbeaten on seven points apiece in La Liga, three behind leaders Real Madrid.