Reds boss Jurgen Klopp confirmed the diagnosis following the game, telling Sky Sports that it was "definitely broken".
The Greece international was forced off the field in the 35th minute after a shoulder barge on the touchline by Bukayo Saka sent him flying off the pitch and into Klopp himself.
In the process, the Greek landed awkwardly, visibly wincing in pain as he clutched his shoulder area.
The Liverpool medical team didn't hesitate or attempt to treat the player on the pitch and he was immediately replaced by Joe Gomez and taken straight to hospital.
The German admitted that it was a big blow for his side with Tsimikas set to be out "for a long time".
Few left-back options remaining
With Andrew Robertson already missing for the Reds having undergone surgery on a dislocated shoulder, this latest injury could be a major blow for Liverpool at this very busy time of year.
Gomez, who replaced Tsimikas on the day, has played at left-back on many occasions but, as a right-sided player, it's not his most natural position.
Nineteen-year-old Luke Chambers, a natural left-footer, is the next player in the pecking order for the role but is unlikely to be risked as a starting player at such an early stage of his development.
That said, he will almost certainly be competing for a place on the bench with 18-year-old Calum Scanlon.