Though they dug in for long spells, they could not find a way past the Premier League giants.
Things might have been different had Roarie Deacon's thunderbolt not struck the bar in the second half.
And Doswell has spoken of his pride in his players but admitted he was disappointed Sutton could not get on the scoresheet.
Arsenal's FA Cup player ratings against Sutton United
Mon, February 20, 2017
Express Sport brings you Arsenal's player ratings from their FA Cup clash with non-league side Sutton United
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David Ospina – 6.5
Paul Doswell has paid tribute to his Sutton side
Theo Walcott scored as Arsenal outclassed Sutton at Gander Green Lane
"It was a dream to watch my team play against Arsenal," he told the BBC.
"The quality of the team they put out showed what's at stake. I'm very very proud.
"We were disappointed not to get a goal for the supporters here but the overriding emotion is so proud.
"This was our cup final and these players will go down in history at the end of the day."
Roarie Deacon caused Arsenal problems
Arsenal will now face Lincoln City in the next round.
And Doswell hopes the minnows fare better than his side did.
"Lincoln and Sutton have done our competition very proud,” he added.
"Best wishes to Danny and Nicky at Lincoln. Go and have your day in the sun like we have."