The Gunners, who smashed West Ham 6-0 at the London Stadium last Sunday, moved back to within two points of Liverpool with another stunning win and also made club history.
Martin Odegaard opened the scoring and Arsenal doubled their lead later in the first half when Leandro Trossard was brought down in the penalty area, giving Saka the chance to score from the spot.
Beating James Trafford in the Burnley goal, Saka recorded Arsenal's 24th goal from a set piece so far in this Premier League season, including penalties.
That tally has beaten the previous record set by the Gunners in 2004/05, when they recorded 23 over the course of the campaign.
Saka recorded his second goal of the game during the second-half at Turf Moor, taking his tally for the campaign to 15.
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However Arsenal weren't done yet, with Trossard and Kai Havertz both finding the net to put the Gunners 5-0 up.
It is also the first time the north London side have won their first five matches in a calendar year in their history.
And it was the first time the Gunners have scored at least five goals in back-to-back league away games.
The emphatic nature of Arsenal's last two victories mean their goal difference is now five better than Manchester City, who play Chelsea on Saturday evening.
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A victory for Pep Guardiola's side would see them go level on points with Mikel Arteta's men.
But they will need to win by five goals or more to usurp the Gunners in second place.