The Catalans had made a superb start to the campaign under new coach Ronald Koeman, hammering Villarreal 4-0 in their opening game before pulling off an impressive 3-0 win at Celta Vigo despite having 10 men for the entire second half.
But they met a very different challenge in Sevilla, who finished fourth in the league last season.
"We knew this would be a really difficult game and so it proved, Sevilla are an excellent team and they played very well, they flooded the midfield and we struggled to cope with that but at least we earned a point," Coutinho said.
"It's been difficult for us to play three games in one week but this is just the start, we'll soon get used to this."
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 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.
Sevilla coach Julen Lopetegui felt his side deserved more.
"It's very difficult to leave this ground with a point, but I'm not satisfied because we had our chances," Lopetegui said.
"It was tough because they are a great team but it's a real shame we couldn't have got more from the game. We are playing with a lot of confidence at the moment but we should demand a little more from ourselves."