Lukaku re-joined Chelsea from Inter Milan this summer for a club record fee of £97.5 million and produced a commanding display on his second debut.
Reece James grabbed the second after Lukaku's opener as Chelsea secured consecutive victories at the start of the Premier League.
When asked about his performance, Lukaku told Sky Sports: "Dominant, I would say. I try to improve every time, I have a long way to go but today was good to win. We have to win and keep going, the Premier League is a very competitive league.
"At the end of the day, you want work hard for the team, score and create chances. I have worked on that and this team is very talented. They are the European champions and they want to keep building and I want to keep winning. We have to keep fighting and keep delivering performances like this.
"You saw a lot, not only from me but he team adapted really well and they made life very comfortable for me and I was not stressed coming into the game. I only had one thing on my mind, winning the game and I am delighted for the club and fans."
Arsenal's disappointing start to the campaign continued as Mikel Arteta saw his side lose 2-0 in back-to-back matches.
Arsenal started brightly but it was Chelsea who opened the scoring after 15 minutes when Lukaku outmuscled Pablo Mari to tap in from James' cross.
It was the striker's first goal for the club after making his Chelsea debut nine years ago.
Chelsea dominated the game from then on and grabbed a second after 35 minutes when James was on the end of a flowing move to lash the ball into the top corner.
The hosts had the first meaningful chance in the second half when Saka's deflected effort was brilliantly tipped over the bar by goalkeeper Edouard Mendy.
Arsenal had an even better chance just before the hour mark with Rob Holding heading just wide from close range after a Mari flick-on.
With 12 minutes remaining, Lukaku came close scoring Chelsea's third but his close range header was brilliantly tipped onto the bar by Bernd Leno.
Leno also denied Kai Havertz late on but Chelsea eased to victory to pile on the pressure on London rivals Arsenal.
Chelsea coach Thomas Tuchel said he was very happy with the way Lukaku had settled into the team so quickly.
"He gives something to our game that we did not have so much, like to protect balls and to play very direct in front and to the last line," Tuchel told Sky Sports.
"You cannot start better than with a goal. He almost scored a second one -- it was a big save by Bernd Leno -- and he assisted some half chances and chances. It's a good start and we have to improve from here."