The England man netted two goals is as many minutes to put the result beyond doubt before halftime, with Gerrard having opened the scoring with a header from a corner.
After Luis Suarez had added a fourth on the break, Sturridge was given the opportunity to score a hat trick from the penalty spot but blazed his effort over.
He showed his frustration at missing the spot kick and then at being taken off with around 20 minutes to go, but Rodgers was keen to focus attention on the forward's display.
“It was a brilliant individual performance from Daniel," the Reds boss told reporters. "His two finishes were sublime and he was outstanding in the game. He’s always a threat, he can score in every single game he plays in and normally does. He was outstanding in the game."
Sturridge was clearly upset at being replaced, but insisted after the match that he was annoyed at his own performance rather than his withdrawal, and Rodgers says that it was much ado about nothing.
“He was just still frustrated after the penalty," Rodgers said. "He apologized to the group. You shouldn't disrespect anyone, in particular the supporters and your teammates. But he hasn’t done that, he was just frustrated and was desperate to get on the score sheet again.”
Rodgers was also keen to highlight the influence of Gerrard, after the Liverpool captain scored his eighth goal in Merseyside derbies and offered Sturridge the chance to take the penalty to seal his hat trick.
“His influence in these games is wonderful. He organizes the team and scored a goal. He’s such a generous man. A local guy in a derby, to give away a penalty [to Sturridge] shows the mark of the man that he is. He led the team and was outstanding.”Gerrard himself acknowledged that he was the one who urged Sturridge to take the spot kick.
"I'll take responsibility for that, I've scored a hat trick against Everton and wanted Daniel to experience that feeling," he told BT Sport. "Daniel was magnificent, he absolutely destroyed them first half and the third goal killed Everton, it knocked the stuffing out of them. Then we knew we could hit them on the break and it worked a treat."