Pep Guardiola's Premier League leaders won 2-0 thanks to goals from Kevin De Bruyne and Raheem Sterling, although they were forced to play the second half without Germany winger Sane.
Bennett scythed into the 22-year-old with an ugly lunge at his left ankle. Sane is set to have a scan on Monday, but Guardiola told reporters he expects to be without the player for two to four weeks as his side enter a pivotal stage of their battle for honours on all fronts.
Another rash challenge on teenage substitute Brahim Diaz saw Bennett earn a second yellow card in stoppage time, with Cardiff boss Neil Warnock accusing the ex-Aston Villa full-back of being unprofessional and promising a punishment of extra training.
Bennett did not mention the Brahim challenge when he posted on Twitter later on Sunday, although he expressed regret over the Sane incident.
"Just want to apologise to Leroy Sane for my tackle today," he tweeted. "I tried to stop the counter-attack and completely mistimed my attempt.
"I didn't mean to catch him like I did. I hope it's nothing serious and he recovers quickly."
Guardiola remonstrated furiously with referee Lee Mason at full-time and repeated his calls for officials to protect players from aggressive foul play during his post-match news conference.
City are back in action on Wednesday at home to West Brom in the Premier League, while next month's return to Champions League action at Basel and the Carabao Cup final against Arsenal are now dates in the calendar that will loom over Sane.
The former Schalke forward has excelled under Guardiola this term, with 11 goals and 14 assists across all competitions.