Clarke sparked another famous giant-killing for the League One Bantams at Valley Parade with a third minute shot which deflected into the net off Black Cats defender John O'Shea.
And he was instrumental in his side's dominant display which left angry visiting boss Gus Poyet launching a tirade against both the media and match referee Kevin Friend.
Clarke said: "They (Sunderland) weren't on the ball like I thought they would be - I was getting in acres of space behind their midfield and I was happy to exploit it.
"Our game plan worked a treat
There's a real belief in the whole city that we can go far and if we get a team at home in the next round I think anyone would find it difficult."
Poyet watched in horror as his side were pinned back in their own half for much of a thrilling first period played on a muddy pitch in front of a sell-out crowd of just over 24,000.
James Hanson twice went close to connecting in front of goal and Clarke forced a save from Vito Mannone while the visitors - reeling from their midweek home loss to QPR - rarely threatened.
Poyet's biggest gripe was a penalty claim after Steven Fletcher inexplicably hesitated in the box with the goal at his mercy and Rory McArdle got back with a great saving challenge.
Poyet said: "They found a goal from an action that can happen in any game
Then when we started understanding a way to hurt Bradford, at that moment we needed to score and the referee needed to do his job and he didn't."
Former Sunderland striker Jon Stead took advantage of a mistake by Adam Johnson to squeeze Bradford's second under Mannone in the 61st minute and the shell-shocked visitors never looked likely to force their way back into the game.
While Bradford greeted the final whistle with raucous acclaim, Sunderland's own massed ranks of fans made their feelings known to Poyet, who had expressed his frustration with some elements of the Black Cats support after the midweek loss.
Asked about his relationship with the fans, Poyet retorted: "The problem is you (the media), not me
"I'm not going to get involved any more
If we close Sunderland - if we put a China Wall around the city - it would be fantastic.
"Now when we let you in and you get out and say what you want we've got a problem
I invite every Sunderland fan and people around the club not to listen to any one of you - only to me."
Poyet added that Jermain Defoe had been left out of the squad as a precaution after picking up a minor calf strain.
Meanwhile the BBC were prompted to respond after criticism that the corporation did not pick the Bradford clash as one of their televised matches
Match Of The Day host Gary Lineker took to Twitter to insist the Bantams would have featured had their fifth round opponents been known at the time of the decision.
Sunderland still faced a replay at Fulham at the time and BBC chiefs were clearly concerned they might commit to a game with no Premier League interest.
The presenter tweeted: "The BBC can only show 2 ties in each round and select them pre replays
If they could've waited Bradford would obviously have been chosen."
Bradford fans made their feelings known during the match and the club later tweeted to Lineker: "see you in the next round, don't stand us up this time".
Parkinson certainly hopes that his men get screen tested next time.
He said: "We were disappointed that it wasn't on the TV
If we just talk about the finances it's a quarter of a million pounds which would have meant a lot, but we move on from it.
"The media has done the talking for us and we've had to take it on the chin and make sure we put in a performance
Hopefully they'll come and give us a live game next time."
Source : PA