Jackson had been on the pitch for only four minutes in the second half when he was shown a straight red for a nasty studs-up challenge on Ayrton Lucas, his second red card in four matches.
"I want to say sorry. To the club, the staff, my team mates, and all the fans watching, I let you down," Jackson, who turned 24 on Friday, wrote on Instagram.
"Another red card, and honestly, I'm so angry at myself. I work very hard every day to help the team not to put us in this kind of situation. I still don't fully understand how it happened. But one thing is clear: it wasn't intentional. Just a football moment that went the wrong way.
"No excuses. I take full responsibility. I'll reflect, I'll grow, and I'll come back stronger for the badge and for everyone who believes in me. Sorry. Sorry. Sorry."
Jackson will serve a minimum one-game suspension. Chelsea next face Esperance Tunis on Wednesday.
Jackson was sent off against Newcastle in a Premier League game last month for a similar challenge.
Chelsea manager Enzo Maresca said: "It's a little bit of a bad moment for Nico in this kind of things. Unfortunately now he will be out for a while, the time that we are here."
Chelsea defender Marc Cucurella said that Jackson had apologised to his team mates after the game.
"He's very sad," Cucurella said. "He's a young player with a lot of quality but maybe needs to improve a little bit in these things."
Check out the match stats from Chelsea's defeat here.
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