Wenger made seven changes from the side thrashed 5-1 at Bayern Munich in the first leg of their Champions League last-16 tie - and his players had enough to see off a team 105 places below them on English football's ladder.
Lucas Perez's cross-shot gave Arsenal the lead after 26 minutes and Theo Walcott doubled the advantage from close range 10 minutes after the break with his 100th goal for the club.
Victory set up a home quarter-final with another National League side, Lincoln City, who beat Burnley on Saturday.
Sutton had their moments, particularly when Adam May wasted a first-half chance from keeper David Ospina's poor clearance, and Roarie Deacon's fierce 25-yard drive struck the bar in the second half.
The result may have gone against them but the hosts emerged from this tie, and this FA Cup run, with huge credit.
Arsenal were on a hiding to nothing after a turbulent week in the wake of their Champions League mauling in Munich, which leaves them on the brink of elimination in the last 16 once more.
The Gunners walked out here with speculation mounting over the future of Wenger and familiar questions being asked about Arsenal's stomach for the fight when the season reaches its pressure points.
Their performance was uncertain and hardly designed to banish the criticism, although allowances must be made for a tricky artificial surface that was heavily saturated before kick-off and again at half-time.
It was simply a question of getting the job done and avoiding embarrassment. There was never going to be any credit in this for Arsenal. And on that basis this can be judged a satisfactory night.
Wenger's troubles were illustrated by the swarm of photographers that surrounded his dugout when he made his entrance - usually the sign of a manager under scrutiny.
The Frenchman, like his players, just needed to get out of Gander Green Lane unscathed and not fall victim to any further humiliation after the harrowing encounter in Munich's Allianz Arena.
This was not a sparkling Arsenal show but they now have what looks like an inviting path to Wembley.
Lincoln may have ousted Burnley, but it takes a huge leap of the imagination to see them denying Arsenal and Wenger a place in the FA Cup semi-finals.
Arsenal still have the chance to add to their tally of 12 FA Cup wins - and Wenger to his total of six.