The officer was injured while arresting a man on Anfield Road after the match and was taken to hospital for treatment before later recovering at home, police said.
The arrests included a man suspected of two offences of assaulting a constable, five people on suspicion of tragedy chanting, one person on suspicion of being drunk and disorderly and assault, two for affray and another suspected of throwing items onto the pitch.
"Officers were on duty outside the ground to smoothly facilitate the movements of those travelling to and from the game, as well as minimise any disruption on the roads outside," Chief Inspector Leanne Hobin said.
"The officers and stewards are there to protect the public and keep them safe, and it is extremely disappointing that an officer has sustained a serious injury while carrying out that duty."
The match at Anfield was played against the backdrop of heightened tensions surrounding Manchester City following a Premier League ruling that found the club guilty of 114 breaches of the competition's financial regulations.
Liverpool supporters made their feelings clear before kickoff, loudly booing City's team announcement and chanting "Going down, going down!" in reference to the possibility of a relegation punishment if sanctions are ultimately imposed.
City deny wrongdoing and have appealed the verdict.