The Chile international had not featured from the outset since suffering a hamstring injury at Norwich City, but set up one and scored the other either side of Sam Vokes' equaliser as Arsenal made it 14 consecutive victories in this competition.
Arsene Wenger made nine changes from last weekend's 1-0 defeat to Chelsea, including a debut for Mohamed Elneny, but it was the creative brilliance of familiar face Sanchez that led to a stunning opener for Calum Chambers, whose only other Arsenal goal came in a 3-0 English Premier League win over the same opponents in November 2014.
Despite promotion from the Championship being their undoubted priority, Burnley only showed two alterations and it was one of those, right-back Tendayi Darikwa, who crossed for Vokes to equalise with a header.
Sanchez had proved a constant threat throughout the first half and he was soon in the thick of the action early in the second period with a composed finish from Alex Oxlade-Chamberlain's delivery.
And there was further good news for Arsenal, who have lifted this trophy in each of the past two seasons, as Tomas Rosicky made his first appearance since May after a knee injury.
Both sides showed attacking intent from the outset, Oxlade-Chamberlain and Sanchez firing narrowly off target for Arsenal before Burnley wasted a glorious opening in the 10th minute.
Top scorer Andre Gray outmuscled Laurent Koscielny and bore down on goal, but his attempted chip was held by David Ospina with striker partner Vokes screaming for the ball to be squared for a simple tap-in.
Seemingly stung into action by that escape, Arsenal camped themselves on the edge of the Burnley box and took the lead in the 19th minute.
Sanchez found himself surrounded by three defenders, but slipped a cute pass into the path of Chambers, who sent a superb finish beyond Tom Heaton into the far corner with the outside of his right foot.
Arsenal continued to look dangerous every time they went forward, but the visitors levelled on the half-hour mark when Darikwa's centre from the right was met by Vokes' powerful header past Ospina from six yards.
But Wenger's side restored their advantage in the 53rd minute when a fine counter-attack released Oxlade-Chamberlain down the right and his cross was swept home by Sanchez.
The holders almost made it 3-1 two minutes later when Koscielny's effort at the back post was headed off the line by Scott Arfield.
Heaton clawed Sanchez's dipping free-kick away from the top corner in the 71st minute as Arsenal dominated, and the Chilean was afforded a resounding reception for his match-winning display when he was replaced by Theo Walcott with 12 minutes remaining.
Walcott had the chance to add gloss to the scoreline in the final minute of injury time when he went clear on goal only to be denied by a sprawling Heaton, but it mattered little as Arsenal's quest for a third straight FA Cup continued.