German defender Volz drilled home a low shot after only 12 minutes at Craven Cottage and Radzinksi coolly converted an Andy Cole throughball in the second-half.
Watford had plenty of the ball and were never out of it until the final whistle. But despite bringing on three subs late on they couldn't find a way back.
It was clear from the start that Fulham manager Chris Coleman was not about to make the same mistake as Liverpool's Rafael Benitez by taking the FA Cup lightly.
With Cole back from suspension he named his strongest possible side, with only injured Steed Malbranque missing.
Watford gave striker Danny Webber the chance to make his first start since November and he was lively, but Fulham produced a much-improved defensive display.
They went ahead when Papa Bouba Diop's ball inside full-back Jermaine Darlington allowed Volz to drill a low shot across veteran keeper Alec Chamberlain and into the net.
Webber ran at the home defence and unleashed a shot that Edwin van der Sar did well to smother after that but Fulham were largely in control.
They almost went 2-0 up after 26 minutes when Luis Boa Morte curled a ferocious free-kick only narrowly wide.
Then they effectively sealed victory after 64 minutes when Cole's superb slide-rule throughball beat the Watford defence and Radzinski raced on to calmly side-foot past Chamberlain.
Watford responded by bringing on Bruce Dyer, Brynjar Gunnarsson and Neal Ardley in a desperate bid to save the game. But it was too late.