Arsenal were poor in the first half at Emirates Stadium, but two goals in the space of three minutes from Saka and substitute Nicolas Pepe secured their third top-flight win out of four in 2020-21.
David McGoldrick gave Chris Wilder's side hope with their first Premier League goal of the season seven minutes from time, but Arsenal held on.
The Blades have lost all four Premier League matches and will need Rhian Brewster, who was not signed in time to play in north London, to hit the ground running after the international break.
Arsenal finally produced the quality they had been lacking to take the lead just after the hour mark, Saka finishing off a slick passing move by nodding Hector Bellerin's cross from the right into the top corner at the back post.
Pepe added a second just six minutes after coming on, cutting in from the right after playing a one-two with Bellerin and clinically picking his spot in the far corner with his left foot.
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McGoldrick curled home from 20 yards out to set up a tense finale and the striker flashed a shot wide in the closing stages with Ramsdale also sent up, but Arsenal were not to be denied all three points.
This article was most recently revised and updated 3 years ago
