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Kai Havertz’s close-range finish in the 16th minute meant the reigning champions left Merseyside with the points.
It was Arsenal’s first win in Anfield over Liverpool in 14 years.
The Reds, who came closest to equalising when Florian Wirtz struck the woodwork, suffered their second defeat under Andoni Iraola.
Trey Nyoni was handed his full Premier League debut as one of seven changes from the Carabao Cup win over Chelsea in midweek.
Victor Munoz and Dominik Szoboszlai returned from injury to begin on the bench.
There was a good pace and even feel to the contest, though a clear-cut chance did not arrive until Havertz’s opener.
After Arsenal won back possession as Liverpool sought to play out from a goal-kick, Declan Rice overlapped Noni Madueke down the right channel and his low delivery found Havertz free at the back post to tap the Gunners into a 1-0 lead.
Liverpool nearly levelled soon after the restart but Bradley Barcola’s threatening cross was an inch or two beyond the reach of Alexander Isak.
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Arsenal then went close to doubling their advantage on three occasions, firstly through a Bruno Guimaraes hit that curled just wide.
Jurrien Timber was then unable to divert Madueke’s knock-back towards goal from close range and Rice forced Alisson Becker into a save on the follow-up.
Taking into account a succession of unsuccessful penalty appeals, the Reds finished the half strongly.
Wirtz whipped a free-kick from the left onto the crossbar and Alexis Mac Allister was frustrated after heading Milos Kerkez’s cross over, as the hosts went into the break behind.
Isak, slipped in by Kerkez, fired over from a tight angle before Szoboszlai was introduced to the affair.
To keep the deficit at one, Liverpool needed vital interceptions from Jeremy Jacquet and Virgil van Dijk and then a stunning Alisson save to keep out Christos Tzolis’ first-time shot.
Up the other end, Ronald Araujo’s shot was blocked by last-ditch actions of Arsenal defenders after Van Dijk headed back into a threatening area.
Munoz got his head onto Kostas Tsimikas’ cross but it was not powerful enough to beat David Raya, and that proved to be the Reds’ last opportunity to equalise.
