The win marked Chelsea's 600th in the league with only United the other team to hit over the 600 mark. The Manchester club have 690 wins.
"It's my dream since I was 11 [to score here]. I've worked hard for this moment. I'm very happy for the win. It was an important game for us. We showed the will to win from the first minute and I'm very happy with the result," Lukaku said after the match.
"In Italy, I was playing in a league where I maybe had one or two chances a game, so I had to take them. I'm very happy with the situation I'm in and we keep going.
"Villa were very good and pressed hard, especially us three up front. They made life very difficult for us. It's great. It's a childhood dream for me. I'm very happy in this situation, but the work has to keep going."
Villa performed well in the first half but several excellent saves from Edouard Mendy meant they failed to register any goals.
"It was very important he [Lukaku] opened up the game for us with the first and scored a fantastic second one," Chelsea manager Thomas Tuchel said.
"He loves these kind of challenges. He was the key factor today."
Villa started the second half strongly with Mendy forced into another good save from Tuanzebe.
They were left to rue their missed chances when Kovacic doubled Chelsea's lead on 49 minutes.
Tyrone Mings attempted to pass the ball back to Steer but hit it short. Kovacic stole it and hit a shot into the bottom left corner.
Lukaku sealed the win on 93 minutes after Cesar Azpilicueta went on a run down the right. He found Lukaku who struck a left-footed shot from outside the box into the net.
"I was pleased with our performance up until the second goal, but we didn't take our chances. That took the wind out of our sails and we allowed a team with the quality they have got to open up the pitch and play keep-ball," Villa maanger Dean Smith said.
"I felt it was an unjust scoreline but we've got a better squad of players this season and the depth and talent we've got allows us to play in Chelsea's half and that's what we did.
"I believe in this group of players that we've got and we can go toe-to-toe with these teams now whereas two years ago we weren't able to."