A 2-0 win for the League One side over their Premier League visitors was secured by an early John O’Shea own goal and Jon Stead’s strike against his former club.
“It was thoroughly deserved. The boys have performed fantastically well. We limited Sunderland to hardly any chances,” said Williams.
Valley Parade was packed out for the fifth-round clash, after Bradford’s win over Chelsea last month, and the celebrations were under way even before the final whistle.
“You can see the scenes here. It’s fantastic for everyone involved,” said Williams.
“We’ve performed fantastically well - the boys don’t need any encouragement. We’re playing against brilliant opposition and they’re all up for it. A full packed house and everyone’s up for the game.
“From the first whistle everyone ran their socks off, worked hard for each other, and it’s thoroughly deserved.
“With the form we’re in at the moment well take anyone at home (in the quarter-finals).
“Being in this position, in the quarter-finals of the cup, it’s a massive reward and bonus for everyone. We’re just living the dream at the moment.”
Stead added: “We’ve done it again and we’re obviously buzzing.”
Billy Clarke’s deflected third-minute effort - which goes down as an O’Shea own goal - and a cool 61st-minute finish from Stead saw the Bantams build on their stunning fourth-round win over Chelsea by dispatching Gus Poyet’s Sunderland with embarrassing ease.
From Clarke’s effort on, the underdogs seized control of the game in front of a sell-out crowd of over 24,000 and heaped more woe on a Black Cats side already reeling from Tuesday night’s dismal top-flight loss to QPR.
At times it seemed like throwback football - a frenetic pace, a muddy pitch cutting up more by the minute, a pair of players swathed in thick white headbands after an early head clash, and a controversial lack of television cameras present to show the action live.
The roar released by the home fans at the final whistle could probably be heard in the offices of TV executives across the land as Phil Parkinson’s men confirmed their richly deserved place in the last eight for the first time since 1976.
The latest chapter in the Bantams’ extraordinary recent tale of cup success began with just three minutes on the clock as a free-kick from Filipe Morais was only half-cleared to Clarke whose lashed effort was deflected by hapless Sunderland captain O’Shea into the net.
The massed ranks of Sunderland fans who were accommodated down one entire side of the ground might have feared the worst when they arrived at the ground to discover recent saviour Jermain Defoe had been left out of the matchday squad.
And while, Defoe apart, the seven changes made by Poyet might have been construed as a team-strengthening exercise in light of their dismal midweek exploits, the hosts were in no mood to be fazed by the big occasion in light of their extraordinary success at Stamford Bridge.
Bradford were simply rampant in the early stages. Clarke had already dug a shot wide before he sent in his effort for the opener, and James Hanson went agonisingly close to connecting with a Stead flick-on to make it 2-0 with barely a quarter of an hour played.
Sunderland belatedly stirred, Adam Johnson bringing the first save out of Bradford keeper Ben Williams and then Steven Fletcher inexplicably hesitating with the goal at his mercy and the bandage-swathed Rory McArdle getting back with a brilliant tackle to preserve the home side’s lead.
Sebastian Larsson had a shot deflected narrowly over the bar in the 21st minute but the danger was mostly coming from Bradford, their sweeping counter-attacks making the Black Cats stars look sluggish and clueless.
Clarke brushed aside Johnson and Billy Jones to bring a save out of visiting keeper Vito Mannone, then Hanson again went inches away from connecting with a Morais cross as the home side ended the first period well on top.
Any home fears the break could work in favour of shell-shocked Sunderland proved unfounded as the end-to-end action continued to swing mostly in Bradford’s favour in the early stages of the second half.
And it was no real surprise when Stead maintained his remarkable record of scoring in every round of this season’s competition as he squeezed an effort under Mannone after being gifted his chance following a mistake by Johnson.
The home side played out the final half-hour of the game with the minimum of fuss, to the extent that by the time referee Kevin Friend blew the final whistle, the home fans had already started celebrating another great achievement.
While Bradford’s players basked in the richly deserved ovation of their fans, Sunderland slunk off to face the music from Poyet after another humiliation in a season which is quickly turning into another one to forget.