The holders had a first-half scare when Burnley's Sam Vokes cancelled out Calum Chambers' 20th-minute opener, but Sanchez's second-half goal ensured the Premier League side avoided an upset against Championship opposition.
Victory provided Arsene Wenger's side with a timely lift, following their Premier League defeat against Chelsea six days previously.
With Chile forward Sanchez and French midfielder Francis Coquelin emerging unscathed after their first starts since picking up injuries in November and Mohamed Elneny making his Gunners debut, Wenger had every reason to be satisfied.
Arsenal return to league action against Southampton on Tuesday when Wenger will no doubt overhaul his line-up.
But after making nine changes to the team that started against Chelsea, the Arsenal manager could be pleased a number of fringe players got valuable match time, while Alex Iwobi produced a promising display on only his third start for the club.
Iwobi, the 19-year-old Nigeria international, occupied the position behind the striker normally filled by Mesut Ozil, while Elneny 23, slotted in alongside Coquelin in front of the back four.
And both players did enough to suggest they will add to the options available to Wenger during the testing final weeks of the campaign.
The visit of a Burnley side currently lying third in the Championship and bidding to make an immediate return to the Premier League, following last season's relegation was never likely to provide Arsenal with an easy passage into the fifth round.
But Wenger's side started positively, making light of the wholesale changes with Sanchez in particular featuring prominently.
- Dominating -
Further back, however, they were less impressive with Laurent Koscielny in particular troubled by the strength and pace of Burnley forward Andre Gray.
And Gray should have put the visitors ahead in the 11th minute when he dispossessed Koscielny and advanced into the penalty area before electing to shoot straight at goalkeeper David Ospina when Vokes was positioned unmarked by the penalty spot.
That was a let off for Arsenal and they quickly punished Burnley for their wastefulness when Chambers put the holders ahead in the 20th minute.
The England defender's only other Arsenal goal also came against Burnley in 2014, but his finish, a well-placed shot struck with the outside of his right boot after being set up by a clever pass by Sanchez, was worthy of a more regular goalscorer.
With Arsenal dominating possession, that should have been the signal for Wenger's side to put the game out of reach.
But Burnley responded positively, levelling 10 minutes later when Vokes headed home from Tendayi Darikwa's cross after the home side twice failed to clear the danger following a corner.
Sean Dyche's team maintained their momentum at the start of the second half, with only a late intervention from Iwobi preventing Scott Arfield from giving the visitors a shock lead.
That appeared to jolt Arsenal into life with the home side reasserting control in the 54th minute when they produced a second goal from a superbly-executed breakaway.
Iwobi was again at the heart of the move, moving the ball on quickly deep inside his own half before eventually picking out Alex Oxlade-Chamberlain, who pulled the ball back into the path of Sanchez.
The former Barcelona star made no mistake as he slotted in a first time finish for his first domestic goal since November, effectively ending Burnley's hopes of springing an upset.