Quins survived a late charge from perennial finalists Leicester to win 30-23 and secure a historic day in the London club's history.
It capped a dominant season in which Conor O'Shea's side had led the Premiership table since mid-September.
Despite that some have suggested Quins have held a false position, after the likes of Leicester's star names spent the initial part of the season on World Cup duty, but England skipper Robshaw believes those doubters have been proved incorrect.
"Some people wrote us off and said we're only there because of the World Cup," he said in an interview on ESPN.
"We've proved them wrong.
"It is so surreal. We owe a lot of it to the fans. It's so nice to repay them."
"It's been about the squad all year, I think this squad has allowed us to fight on all fronts."
Quins looked home and dry when they led 30-13 with 14 minutes left but the Tigers struck back with a converted Anthony Allen try - after a clever Ben Youngs break - and a George Ford penalty to set up tense final 10 minutes.
"At the end it was nerve-wracking but that's what we've been like this season," Robshaw said.
"It was a bit difficult, especially when Ben Youngs went through. They scored off a few of our mistakes today but we held on."
Nick Evans capped his own remarkable season with silverware and added: "What a season. That game sums up our season - we play well, get two scores in front and let them (opponents) back in.
"We've been there to be shot at and we've held them off all season so it's great.
"We've thoroughly deserved this."
Asked if the breakthrough success had erased the club's recent controversies, including the infamous 'Bloodgate' scandal, Evans added: "They're gone. As far as I'm concerned they're gone.
"All due respect to media, it's a tattoo on the club, but those players have left.
"I hope it is (the start of a new chapter). If we stay together as a team then I'm sure we'll get a lot more wins and a lot more trophies hopefully."
O'Shea warned today's win could signal the start of a period of dominance for his side.
"A couple of years ago this team said it wanted to create a different chapter in Quins history," he said.
"We want to be a great side. This has to be a starting point. We'll start from scratch next year, but we'll enjoy tonight.
"It's a very special day for the club. I'm so proud of the effort of those players."
Leicester director of rugby Richard Cockerill conceded Quins deserved their victory and bemoaned his side's lack of composure.
"We didn't cherish the ball. We tried to force it a bit too much which we didn't want to do," he said.
"They outplayed us at times although we still managed to score two good tries and fight right until the end again.
"But they probably deserved it. We will go away now and regroup."
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