Van der Vaart netted the only goal of the game in the 42nd minute at Vicarage Road with a 30-yard drive that beat home goalkeeper Scott Loach, but Championship side Watford created enough chances to dump their Premier League opponents out.
The Hornets struck the Tottenham woodwork on two occasions and visiting goalkeeper Cudicini performed well to keep out lively striker Marvin Sordell in the second half.
When asked if he was relieved to get the win, Van der Vaart told ESPN: "Yes, I think so. I think Watford deserved more, maybe a win because they were the better team tonight.
"Sometimes you have these types of days, the 100% sharpness isn't there and I think it was a difficult game, the pitch was heavy and they played a great game."
Regarding his goal, which Loach may feel he could have done better with, the Dutchman said: "It was a lucky strike but I'm happy with the win."
Van der Vaart also praised team-mate Cudicini, adding: "(He made) great saves. We were under pressure and he did fantastic."
Cudicini was not surprised that Watford gave them such a tough night, saying: "Once you've been in England for a while you know what the FA Cup is about so we knew it was going to be a difficult game."
Asked about his side's chances of going all the way in the cup, the Italian added: "We have a great squad we showed that in Premier League and hopefully we will have a long run in the competition."
Watford manager Sean Dyche told ESPN: "I thought our performance was excellent tonight.
"Sometimes it happens when you don't get your rewards and I think that happened tonight.
"Performances are really important but I'm really gutted we couldn't go through tonight.
"We should have got a draw as a minimum."
Tottenham manager Harry Redknapp also felt his side rode their luck to go through, heaping praise on Watford in the process.
Redknapp said: "They say it's better to be lucky than good. Tonight we were lucky, we weren't good.
"We're normally good, tonight was not the Tottenham team I see every week, we didn't play well, we were poor.
"Credit to Watford though, they were fantastic. I thought they were absolutely brilliant. They were top class. We got away with it tonight.
"I don't know what was wrong with us, we just never got hold of the ball. We were off colour."
Redknapp added: "Sometimes you need a bit of luck and we got it tonight. When I won the cup with Portsmouth I remember going to Preston and getting absolutely battered for 90 minutes and scoring in the 90th minute and going on to win the cup.
"We're still there and I think we've got a big chance this year of winning the cup, I think this could be our year."
Redknapp also admitted it was unlikely Spurs would be bringing in any new faces before the January transfer window closes next week.
He said: "It's difficult, I doubt if we'll do anything really now. It'll be hard to do anything.
"I haven't been chasing the chairman, we haven't been desperate. The chairman's not been pushing me to do anything. There' s a couple of players that we like but you're not going to get them in at this stage."
