The Dutchman's 25-yard strike eluded Watford goalkeeper Scott Loach to ease the Premier League side's nerves after the Championship side had dominated possession in the first half at Vicarage Road.
Everton joined Spurs in the fifth round with a 2-1 win over Fulham.
Spurs, third in the top flight but without injured winger Gareth Bale, struggled to assert their authority in the second half and Watford had good chances to equalise, John Eustace striking the post with a far-post header.
Spurs manager Harry Redknapp was in the dugout after a week spent at Southwark Crown Court where he is on trial for tax evasion, which he denies.
"They say it's better to be lucky than good and tonight we were lucky," Redknapp told reporters. "That was not the Tottenham side I see every week, we were poor. Watford were fantastic, top class. We rode our luck and we got away with it.
"I don't know what was wrong with us. We were off colour but we got the luck. I think we've got a big chance of winning the cup this year. A win's a win and we are in the hat."
Tottenham, eight-times FA Cup winners, were joined in the fifth round by Everton who came from behind to beat Premier League rivals Fulham at Goodison Park.
Marouane Fellaini squeezed a header past Fulham goalkeeper David Stockdale in the 73rd minute to secure Everton's progress.
Danny Murphy had fired Fulham ahead from the penalty spot in the 14th minute but Argentine striker Denis Stracqualursi levelled 13 minutes later when he headed home Landon Donovan's cross.
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