Martinez, restored to the line-up after missing out on a transfer window exit, saved from Jack Grealish and Michael Keane to ensure Villa avoided a third straight Premier League defeat.
Emery’s team have set an unwanted record by becoming the first Villa side ever to fail to score in their opening four matches of the season.
But Martinez ensured it was not worse. The 33-year-old had been left out of Villa’s previous match, a 3-0 defeat to Crystal Palace, because Emery was concerned about his lack of focus.
Emery said: “I am so, so happy. We need him. We struggled in his situation, but we have to get feelings together and feel the collective objective. Today his comeback has been fantastic.
“We have to protect him and feel him inside the group so he is comfortable and confident. He was fantastic.”
Villa remain 19th in the table and still searching for their first win of the campaign.
An Emi Buendia shot, deflected just wide of the post, was the closest they came to scoring on their first-ever Hill Dickinson Stadium.
They were also indebted to a dreadful miss from Everton forward Beto inside the opening two minutes.
“We are not scoring or winning, but we are building our spirit,” said Emery. “We are getting emotions together and balance.
“For me, today there were a lot of very good things. We need to win, but this point helps us.”