Everton had been indebted to Tim Howard for keeping the scores level at half-time after the American keeper produced a superb save from Christian Benteke's early penalty.
Martinez was satisfied to come away from Villa Park with a clean sheet and he was delighted with the performance of Howard.
Commenting on the second-half performance, Martinez told Sky Sports: "It was very, very impressive because remember you need to give credit to Aston Villa for the way they played the game, they were always on the front foot.
"They are always going to create problems with the way they attack and that opens up the game in the first half, and we could create problems for the Villa back four, and they could create problems.
"The second half was very impressive in the way we dictated the play and we limited Villa's attacking play to very little. That was the biggest satisfaction today, to be able to improve the level in the second half and perform in a manner that was worthy of three points.
"I thought Tim was sensational. We started a little bit slowly and he was there to maintain the standards. We've got him to thank for a very important clean sheet."
Martinez was also impressed with Lukaku, who scored his fifth goal in five Premier League appearances since joining on loan from Chelsea.
The Everton boss added: "Since he arrived, he has never needed a settling-in period and that's credit to everyone at the club, from the fans to the experienced players who have allowed him to enjoy his football.
"The way he took his goal was just real composure and his overall contribution to our build-up play was really impressive, the way he kept the back four occupied.
"He's a young man who is already developing into someone who can fulfil their potential, and that's extremely exciting. We want Rom to carry on enjoying his football."