During the second half, the Uruguayan was caught by TV characters sinking his teeth into the Blues man's arm before later going on to net an injury-time equaliser.
The forward said in a statement: "I am deeply sorry for my inexcusable behaviour during our match against Chelsea.
"I have issued an apology and have tried to contact Branislav Ivanovic to speak to him personally. I apologise also to my manager, playing colleagues and everyone at Liverpool for letting them down."
Manager Brendan Rodgers confirmed that he had reprimanded Suarez over the incident, explaining: "Having reviewed the video footage and spoken to Luis, his behaviour is unacceptable and I have made him aware of this."
The club's managing director, Ian Ayre, also made clear his disappointment with the incident and indicated that the club would accept any punishment from the Football Association which may be forthcoming.
"Luis has made an unreserved apology for his actions today," wrote Ayre.
"His behaviour is not befitting of any player wearing a Liverpool shirt and Luis is aware that he has let himself and everyone associated with the club down. We will deal with the matter internally and await any action from the FA."
It was an eventful game for Suarez, who had already conceded a penalty earlier in the second half for a handball, which Eden Hazard scored.
Following the match, Chelsea goalkeeper Petr Cech had disclosed that Ivanovic was "not happy at all" about the incident.
