Barca were put under immense pressure from kick-off and fell behind to Vitolo's 15th minute strike.
Messi levelled just before halftime following a slew of missed chances from the hosts. After the break, Barcelona took control and Luis Suarez fired in the winner in the 61st minute.
Barca stay two points behind Real, who lead the standings with 27 points.
Villarreal beat Real Betis 2-0 to move up to third, three points behind Barca. Fifth-placed Sevilla, on 21 points, trail Atletico Madrid on goal difference.
Sunday's meeting between Jorge Sampaoli's relentless Sevilla and a Barcelona side smarting from defeat at Manchester City promised a lot and lived up to the hype as a breathtaking end-to-end encounter ensued.
Sevilla opened the scoring as Luciano Vietto's through ball caught Sergi Roberto unawares and Vitolo swooped in to slot past Ter Stegen.
Sevilla grew in confidence and looked to grab another, Vietto firing wide from a tight angle and Vitolo crashing over the bar.
But Barca drew level following a wonderful break by Messi which drew in Denis Suarez and Neymar before the Argentine finished off.
Sevilla were punctured by the goal and Barca dominated the second half.
Ivan Rakitic and Messi were both denied by Sergio Rico but Suarez showed more composure, sliding the ball through the goalkeeper's legs and ending Sevilla's 100 percent record at home in the league and Europe.
"Comebacks like this always make a team stronger, especially when you remember no-one else has won at this ground," said Barca coach Luis Enrique.
"It was a wonderful game for attacking players, who wanted the ball and wanted to take risks. It made for a hectic game. They took the lead then we found solutions. I'm very happy with how the day has gone."
Earlier on Sunday, Madrid got off to a slow start against neighbours Leganes, but Bale put them in charge in the 37th minute.
The Wales international raced onto a through ball from Isco that split the visiting defence and rounded goalkeeper Jon Serantes to tuck the ball inside the near post from a tight angle.
Bale followed on with his 150th career goal, cashing in on poor defending from a free kick.
Morata rounded off the win in the 76th minute as Real preserved their unbeaten run in all competitions and also kept a first clean sheet in 11 games.
"We keep picking up points and today we did it while playing well, even if at the start we gave the ball away a lot. When we found the first goal the game changed," said Zidane.
"We bossed the second half and I'm happy with our performance and to have kept a clean sheet. We're not going to win the title with 100 points, because it's becoming much harder to win games in this league."
Villarreal kept up their excellent start to the season by beating struggling Real Betis thanks to two scorching long range strikes, one from Manu Trigueros and the other from Roberto Soriano either side of halftime.
Betis have lost four of their last five games, with pressure mounting on coach Gustavo Poyet, who started the game with top scorer Ruben Castro on the bench.
Elsewhere, Celta Vigo came from behind to beat Valencia 2-1 and Espanyol drew 0-0 at home to Athletic Bilbao.
(Reporting by Richard Martin; Editing by Pritha Sarkar and Ian Ransom)
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