Europa League holders Sevilla took an early lead when Dutch forward Luuk de Jong pounced on a poor headed clearance in the area and smashed into the net, flummoxing Barca keeper Neto.
But the Catalans hit back less than two minutes later when club record signing Coutinho, back at the club after spending last year on loan with Bayern Munich, reacted fastest to fire home after a Lionel Messi pass was deflected into his path.
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.
Barca and Sevilla each have seven points after three games and are fifth and sixth in the standings, with champions Real Madrid top on 10.
The Catalans had made a superb start to the campaign under new coach 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."
"It's very difficult to leave this ground with a point but I'm not satisfied because we had our chances," he 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."
Barca's new signing Sergino Dest made his debut three days after completing his switch from Ajax Amsterdam, coming off the bench to replace the injured Jordi Alba.