The two sides could have scored more, though, as Arsenal hit the woodwork twice and Chelsea struck the post as well.
The second of Arsenal’s efforts that hit the woodwork came from Joe Willock late on and replays suggested his strike, that hit the crossbar, crossed the line.
The goal was not given, though, and Chelsea held on to claim the win, while Arsenal's afternoon was made worse by Thomas Partey going off injured.
Both teams named strong sides for this opening of the Mind Series, with Arsenal also including new signing Ben White on the bench.
Chelsea boasted a strong bench too as N’Golo Kante was among their substitutes after returning from his holiday post Euro 2020.
Arsenal started the game brightly and they were getting joy down the left as Kieran Tierney and Pierre-Emerick Aubameyang linked up well.
It was Chelsea, however, who took the lead in the 26th minute as they hit Arsenal on the counter. Timo Werner was able to burst through and, with only Pablo Mari back defending, he played in Havertz to score.
Arsenal tried to hit back after that and they almost equalised, but Emile Smith Rowe’s shot from outside the box hit the post.
Chelsea then hit the woodwork themselves through Hakim Ziyech - while Callum Hudson-Odoi drew a good save from Bernd Leno before half-time too - however Arsenal’s biggest concern was Partey going off injured.
The midfielder appeared to be struggling with an ankle issue and the sight of him limping off will worry Mikel Arteta given the Gunners’ Premier League opener with Brentford is on August 13.
White came on at the break for his unofficial debut and it wasn’t long before he was making a name for himself as he produced a brilliant goal-line block to deny Ziyech.
At the other end, Arsenal were still searching for an equaliser and they went close as Edouard Mendy got done low to save Alexandre Lacazette’s free-kick.
They finally got back on level terms with 23 minutes to go as Xhaka, who had come on for Partey, rose highest to head home Nicolas Pepe’s corner.
It was to be short-lived, though, as just moments later Chelsea were back in front when Hector Bellerin’s stray pass allowed Abraham to score.
Arsenal pushed hard for an equaliser and they will feel they should have got one. First, Aubameyang missed a great chance inside the box and then Willock hit the bar.
Willock’s effort appeared to be over the line too but, without goal-line technology in use, the officials did not give it and the youngster was denied the chance to level the game.