Gerrard takes bragging rights against former teammate Lampard in first managerial clash
Both teams fell to opening day defeats last weekend, but Gerrard's side turned their fortunes around at the expense of the visitors under Lampard, his former midfield rival when they played for Liverpool and Chelsea respectively.
"I'm really pleased with our level of performance, I believed that if we found that level there would be a different outcome from last week," Gerrard told the BBC.
"We looked a different team today. It was tough conditions for both sets of players, but they were excellent in terms of the game plan. We really wanted to make it about their back three, and both goals have come off that."
Villa's Ollie Watkins managed to stretch the Everton defence as he created goals for Danny Ings and Emiliano Buendia and, despite a late own goal by former Everton player Lucas Digne, Villa stood firm to claim the three points.
"I thought we should have got a draw at the end of the game, we had two or three good chances late on. When we were set defensively and organised, I didn't think they gave us huge problems," a disappointed Lampard told BT Sport.
"When we gave the ball away, a couple of transitions, they scored two goals. That's the reality of Premier League football, you can't make those mistakes and give the ball away when you're open, we got punished for that."
The defeat was Everton's second in their opening two games of the season and they face newcomers Nottingham Forrest next Saturday at Goodison Park, while Villa travel to Selhurst Park to take on Crystal Palace.