The Bundesliga giants struggled to get going in the first leg of the quarter-final and were fortunate not to concede before Pablo Sarabia 32nd-minute opener on Tuesday.
However, they were gifted an equaliser just five minutes later when Sevilla's Jesus Navas turned Franck Ribery's low cross into his own net.
There was an element of good fortune about their winner too, as Thiago's header beat goalkeeper Davis Soria via a deflection off team-mate Sergio Escudero.
Heynckes is expecting better from his players as the competition moves forward.
"We lost the ball too often and lacked structure in midfield, which allowed Sevilla to create chances," he told the media.
"We improved in the second half and in the end deserved to win.
"Psychologically, it was very important to equalise quickly, even if it took a bit of luck. But if we want to win the Champions League, we need to improve."
While happy enough with the result at a raucous Ramon Sanchez Pizjuan, Bayern defender Mats Hummels admitted they had been second best to Sevilla before the break.
Still, the turnaround puts the German club in control of the tie ahead of the return leg at the Allianz Arena next Wednesday.
"We would have signed off on that result if you’d offered it to us before the game," Hummels told BT Sport.
"But it wasn’t 90 good minutes from us. In the first half Sevilla were the better team and had chances to score a second goal. We know there's a lot to improve on.
"We didn’t want them to be that dominant for the bigger part of the first half, that's something we will analyse and talk about to make sure it doesn't happen again."