Dominic Calvert-Lewin scored once again to become Everton's first player to score in the opening four matches of a Premier League season.
Calvert-Lewin took his tally to nine goals from six games in all competitions when he rose at the far post to head home a Gylfi Sigurdsson cross in the 16th minute before Neil Maupay levelled from close range in the 41st minute.
Yerry Mina restored the home side's lead on the stroke of halftime, heading home an inch-perfect free kick floated in by fellow Colombian Rodriguez.
Good work from substitute Alex Iwobi presented Rodriguez with a second-half brace as he swept the ball home from eight metres to make it 3-1 in the 52nd minute before he capped another flowing move in the 70th minute.
Midfielder Yves Bissouma pulled one back for the visitors with a cracking shot from long range in stoppage time but it was too little, too late to deny high-flying Everton a fourth successive league win.
Calvert-Lewin acknowledged he was full of confidence and added that the performance, although it left something to be desired, showed a new level of collective and personal maturity.
"I think it's about self-belief," he told BT Sport. "I always believed that I could get here. Sometimes it's about age and maturity. I've been patient, now I'm reaping my rewards.
"I think today, particularly in the second half, we didn't play that well. The best teams win when they're not always on top. We're confident, creating chances and scoring goals."