Pedro Neto's double on Saturday at the Hill Dickinson Stadium ended Everton's unbeaten start to the league season, but Chelsea's victory was overshadowed by the lengthy injury stoppage in the second half.
Martinez and Branthwaite came together when Everton's Kiernan Dewsbury-Hall whipped a free kick into the box and, although the defender was back on his feet after treatment, the Argentina goalkeeper required treatment from paramedics.
The 34-year-old was eventually strapped to a stretcher and carried off the field before being taken to hospital with a concussion, as Argentine media reported he had suffered a jaw injury in the collision.
Martinez, wearing a hospital gown, posted on social media early on Sunday: "Thanks to everyone for the messages. The operation went well.
"The most important thing is that we won and kept a clean sheet. In a few weeks we'll be back."
Martinez's injury forced Chelsea manager Xabi Alonso to give Mike Penders his Premier League debut, and the 21-year-old keeper helped Chelsea keep their first clean sheet in the league since January.
Penders, a Belgian youth international, will likely have to step in until Martinez is cleared to return to training.
Chelsea host Tottenham Hotspur in the league next Saturday followed by a League Cup fourth-round game at Liverpool on October 28 and a home Premier League match against Manchester United three days later.
"Mike has personality. He's brave and we have full trust with him. He did well," Alonso told reporters.
"He played in the cup games and whenever he's needed, like the other players whenever they are needed they can perform.
"It's all about the squad, that all the players feel important and can have their moment whenever they are needed, and that's very positive."
