The Chile international capitalised on two errors from the hosts to take his tally to five goals in eight Premier League appearances this season
Arsene Wenger’s men have now recorded three wins in the Premier League this season and move into fifth place, eight points behind leaders Chelsea who travel to Manchester United on Sunday.
Sunderland headed into the fixture off the back of an 8-0 hammering at the hands of Southampton last Saturday and another two lapses in defence were responsible for their third league defeat of the campaign.
In the 30th minute, Wes Brown attempted to control a lofted ball forward from Per Mertesacker but his back-pass to Vito Mannone was under-hit and it allowed Alexis to steal in and easily finish.
The Chile international netted his second in injury time after Mannone failed to control Patrick van Aanholt's back-pass, which presented the Arsenal winger with an empty goal.
Arsene Wenger, without the suspended Jack Wilshere, made three changes, with Wojciech Szczesny returning in goal having been suspended in midweek and Alex Oxlade-Chamberlain and Mikel Arteta given starts.
That meant Emiliano Martinez and Aaron Ramsey had to settle for places on the bench, where they were joined by Theo Walcott - who was appearing for the first time having spent nine months on the sidelines with a cruciate injury.
Arsenal's crisp play led to a first opening after 15 minutes, Danny Welbeck cutting in from the left and sending a powerful curler just over.
The home side enjoyed a good spell afterwards, only to go behind due to an individual error after 30 minutes.
Brown was the culprit, mis-hitting an attempted back-pass to allow Sanchez to race through and delicately chip Mannone for his fourth league goal since joining from Barcelona.
Not content with one, Arsenal continued to press forward with Santi Cazorla shooting just wide before Calum Chambers forced Mannone into a save.
Sunderland were dealt a blow early in the second period as injury forced off Steven Fletcher, Wickham deputising.
Wickham's introduction did not have much impact initially as Arsenal picked up where they left off, epitomised by Sanchez drawing a save from Mannone with a strike from distance.
A feistier spell ensued after that effort as Sunderland attempted to rally, Sebastian Larsson stinging the hands of Szczesny with a powerful 20-yard attempt.
That proved just a brief flurry, however, as Arsenal quickly reassumed control - albeit without really threatening a second goal.
Arsenal almost added another with 12 minutes to go, only for Welbeck to head tamely over after being picked out by the marauding Nacho Monreal.
Welbeck's miss set up a nervy ending to the game, with Van Aanholt volleying wide from distance after Szczesny had left his line.
Cazorla missed a great chance to add a second when firing just over, but it mattered little as Sunderland's vulnerability reared its head again in the 92nd minute following Mannone's blunder which allowed Sanchez to seal the three points for Arsenal.