Manchester City and Tottenham Hotspur also progressed, but West Bromwich Albion were taken to a replay by third-tier Peterborough United and Bournemouth had to come from behind to beat 2008 FA Cup winners Portsmouth.
Arsenal’s win at the Emirates Stadium preserved their hopes of becoming the first team to win three consecutive FA Cups since Blackburn Rovers, who achieved the feat between 1884 and 1886.
Making his first start after two months out with a hamstring injury, Chile star Sanchez teed up Calum Chambers to open the scoring in the 19th minute with a fine shot using the outside of his right foot.
Sam Vokes equalised for Burnley on the half hour with a powerful header, but Arsenal, who made nine changes, claimed victory in the 53rd minute when Sanchez slammed home after a neat team move.
Nigerian starlet Kelechi Iheanacho scored a hat-trick as City kept alive their dream of an unprecedented quadruple of trophies by easing to a 4-0 win at Aston Villa, last season’s beaten finalists.
Iheanacho opened the scoring in the fourth minute, following in to score after Fernando’s header was cleared off the line, and the 19-year-old forward added a second with a 24th-minute penalty.
He completed his hat-trick with 16 minutes remaining, seizing on ex-City defender Micah Richards’s back-pass and surging through to score, before Raheem Sterling completed the scoring.
Manuel Pellegrini’s City are second in the Premier League, face Dynamo Kiev in the Champions League last 16 and will tackle Liverpool in the League Cup final on February 28.
Liverpool are in action later on Saturday when they host West Ham United, who have beaten them twice in the league this season.
Tottenham breezed through, Nacer Chadli scoring twice on his 100th appearance and Eric Dier and Tom Carroll also finding the net in a 4-1 win at third-tier strugglers Colchester United.
– King, Pugh rescue Bournemouth –
“I think that this competition is magic. The opponents always believe that maybe it’s their day, that they can win,” Spurs manger Mauricio Pochettino told BT Sport.
“At times we invited them a little bit to believe in the possibility, but I think it was always under control. We conceded some chances, but it was a fair result.”
Liverpool beat Stoke City on penalties to reach the League Cup semi-finals on Tuesday and Mark Hughes’s men met with fresh cup disappointment four days later as they went down 1-0 at Crystal Palace.
Wilfried Zaha notched the game’s only goal in the 17th minute at Selhurst Park, taking Palace into round five for the second season running.
Gary Roberts put Portsmouth, now languishing in the fourth tier, within sight of a major upset when he scored in the 43rd minute of their meeting with south-coast rivals Bournemouth.
But Josh King equalised for Eddie Howe’s side with a 71st-minute header before Marc Pugh completed a 2-1 win.
Saido Berahino’s brace was not enough for West Brom, who drew 2-2 with Peterborough, but Odion Ighalo’s late strike earned Watford a 1-0 victory at Championship representatives Nottingham Forest.
Fourth-tier Oxford United could not repeat their third-round win over Swansea City, losing 3-0 at home to Blackburn Rovers, but third-division Shrewsbury Town stunned Championship side Sheffield Wednesday 3-2.
Chelsea interim manager Guus Hiddink’s quest to repeat his 2009 FA Cup triumph with the west London club continues on Sunday when his team visit second-tier Milton Keynes Dons.
Manchester United were the first team to book a place in the last 16 after winning 3-1 at Championship promotion chasers Derby County on Friday, with Wayne Rooney, Daley Blind and Juan Mata scoring.