Manchester City and Tottenham Hotspur also progressed, but West Bromwich Albion were taken to a replay by third-tier Peterborough United and Liverpool drew 0-0 with West Ham United.
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.
"We would love to do it, but there is a long way to go," said Wenger, who gave a debut to Egyptian midfielder Mohamed Elneny and also welcomed back holding midfielder Francis Coquelin.
"For the first 20 minutes we looked quite in control, but when they equalised suddenly you could see energy, commitment, and we had to dig deep to win the game."
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 Manchester 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 Idrissa Gueye's calamitous back-pass and surging through to score, before Raheem Sterling added a fourth.
"Kelechi is a very good young player," said City manager Manuel Pellegrini. "We let (Stevan) Jovetic and (Edin) Dzeko go because I see Kelechi every day and know what he can do."
- Bournemouth thwart Portsmouth -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.
Jurgen Klopp's Liverpool were taken to an unwelcome replay by West Ham, who had previously beaten them twice in the league.
Showing 10 changes, Liverpool dominated proceedings at Anfield, but could find no way past West Ham stand-in goalkeeper Darren Randolph.
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.
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 lost 1-0 at Crystal Palace, with Wilfried Zaha scoring.
Portsmouth, FA Cup winners in 2008 but now languishing in the fourth division, were in sight of a major upset when Gary Roberts put them ahead 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 secured 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.
First Published On : Jan 31, 2016 12:35 IST