Media playback is not supported on this device
The National League side were beaten 2-0 by Premier League visitors Arsenal, who are 105 places above Sutton on English football's ladder.
The home side had several chances, with Adam May wasting a chance and Roarie Deacon hitting the crossbar.
"This was our cup final," said Doswell. "I'm very, very proud."
The 50-year-old said it was "a dream" to watch his side play Arsenal, who will face another non-league team - Lincoln City - in the last eight.
"We were disappointed not to get a goal for the supporters here but the overriding emotion is pride," he added.
Sutton, who famously beat Coventry in the FA Cup in 1989, had overcome league sides AFC Wimbledon and Leeds United on their way to the fifth round.
But they could not get the better of an Arsenal team who had lost three of their previous four games.
FA Cup fairytale over for Sutton United - 5 live podcast
This content will not work on your device, please check Javascript and cookies are enabled or update your browser
Sutton held their own against 12-time FA Cup winners Arsenal, who are fourth in the Premier League, before Lucas Perez's 26th-minute cross-shot gave Arsene Wenger's side the lead.
Theo Walcott's 100th goal for the club sealed victory at Gander Green Lane.
Wenger, who made seven changes to the team beaten 5-1 by Bayern Munich last week, said Sutton's performance surprised him.
"I don't really enjoy tonight because we absolutely had to do the job and it is tricky," he said.
"They played very well. It is basically division five and when I arrived here in England 20 years ago, in division five they were not as fit physically as they were today.
"They were organised and had a huge desire. If we were not mentally prepared, we would not have gone through."
Former Arsenal defender Martin Keown was surprised to see Deacon, who came through the Gunners' youth system, playing at non-league level.
"You wonder, looking at Roarie Deacon, if the game has failed him," he said.
"He should be playing at a higher level. He has great quality with both feet and he was really unlucky with that shot that hit the bar."
Former England striker Alan Shearer said the Sutton players can be very proud of their performance throughout the competition.
"It has been one heck of a run," he said. "There were some very, very positive performances out there."

Revenue for the first quarter dropped 64% to 14.8bn Taiwanese dollars ($456m; Â£315m), while the net loss was 2.6bn Taiwanese dollars, compared with a profit the year before.
The firm has been struggling to compete with the likes of Apple and Samsung.
It is pinning hopes on its new HTC 10 phone to revive fortunes as well as its virtual reality headset Vive.
"The media and consumer buzz around HTC, including for the keenly-awaited launches of the flagship smartphone and Vive virtual reality system, clearly demonstrate our leadership in innovation and have provided a great boost to the HTC brand," said Cher Wang, chairwoman and chief executive of HTC in a statement.
Chief financial officer Chialin Chang added the company was hoping that the smartphone business would break even in the third quarter.
The HTC 10, which comes with an "ultraselfie" front camera designed to reduce the risk of blurry shots, and the Vive headset were both only launched in April and so are not reflected in this current quarter.
The success of those two products will dictate the success of the company, said analyst Ben Wood from CCS Insight.
"The next quarter will be critical because we'll start to see the effect of Vive and the HTC 10 phone. The company has really got to work hard to show signs of progress," he said.
"The combination of Apple's incredible success and Samsung's determination to outspend all its rivals on marketing puts HTC in a precarious position."
Five years ago, HTC was the world's fourth bestselling smartphone maker with a market share of about 9%.
But in 2015, it fell to 17th place with a share of about 1%, according to research firm IDC.
Sutton United's players will "go down in history" despite missing out on a place in the FA Cup quarter-finals, said manager Paul Doswell.
[NEXT_CONCEPT]
A sharp plunge in sales has pushed the Taiwanese phonemaker HTC into a loss for the three months to March.