Kevin Gameiro scored from the spot seconds after the restart but failed to do it again two minutes later, striking his second penalty against the crossbar.
Sevilla's Argentine Ever Banega made amends with their third spot kick in the 67th minute and Yevhen Konoplyanka wrapped up the win in the 84th with a clever shot from a tight angle that beat goalkeeper Yann Sommer.
Gladbach, who have made their worst start to a Bundesliga campaign with four consecutive defeats, were no match for the speedy Spaniards and struggled to contain them throughout the entire game.
Sevilla, still without a La Liga win this season, should have scored in the fifth minute when Vitolo hit the post with a low drive and captain Jose Antonio Reyes somehow managed to fire over the bar from five metres.
Gladbach responded with a chance of their own but Roel Brouwers' glancing header sailed narrowly wide.
But the Spaniards kept pushing forward and earned a penalty seconds after the restart when Sommer brought down Vitolo. Frenchman Gameiro sent the Swiss keeper the wrong way to put the hosts ahead.
Gladbach, making their first European Cup appearance in more than 37 years, were lucky to see Gameiro then rifle a second spot kick, given for another foul on Vitolo, against the bar before Banega stepped up and kept his cool with their third penalty to seal the victory.
Sevilla, who next visit Juventus and Manchester City, with the Italians having won 2-1 in England on Tuesday, added a third when Konoplyanka caught Sommer off guard and the keeper fell into the goal with the ball.
(Reporting by Karolos Grohmann; editing by Ken Ferris)
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