Sutton more than held their own throughout the 90 minutes but they were behind in the 27th minute after Lucas Perez's strike broke the deadlock.
After the break, Theo Walcott grabbed his 100th Arsenal goal in the 55th minute to ease the Premier League side's nerves.
But Sutton continued to look for a way back into the tie and were inches away from making it an uncomfortable ending for the Gunners, but Roarie Deacon's powerful shot cannoned back off the crossbar.
