It was a slow start to the contest at Vicarage Road with Troy Deeney, Stuart Dallas and Juan Carlos Perdes having the only real opportunities in the first 45 minutes, all failing to capitalise.
Quique Sánchez Flores' side perked up in the second period with Jose Holebas having a great chance before Scott Wootton turned the ball into his own net to give the Hornets the advantage.
The hosts shut up shop from that point, with Deeney, Holebas and the impressive Souleymane Doukara having half-chances, but they had already done enough to reach the last eight.
It didn’t take long for the Premier League side to have their first opportunity of the game when Deeney rose highest in the second minute to a ball in from the right flank, but his header was just a few yards wide of the post.
Despite their early dominance, neither side managed to establish themselves in the match with the next real chance falling to Leeds in the 26th minute when Mirco Anetenucci turned into space on the halfway point, driving at the Watford defence before picking out Dallas, who failed to test Costel Pantilimon.
The last chance of the half came four minutes from the end when Deeney slipped the ball into Paredes, who struck a low and dangerous ball towards the near post but Marco Silvestri did well to gather ball at the second attempt.
United's defence were called upon massively at the beginning of the second period when they were forced into making two crucial blocks from Deeney who pleaded for a penalty but to no avail as Michael Oliver waved away the appeals.
It was yet another block seconds later when the ball fell for Holebas at the back post but yet another excellent block prevented the Hornets taking the lead.
Despite their excellent defending early on, Steve Evans' side could do nothing to prevent them going behind in the 54th minute.
Watson was causing problems once again as he whipped a dangerous ball towards the back post, from which Wootton, who was under little pressure, managed to turn the ball into his own net to give the Premier League opposition the lead.
Etienne Capoue thought he had made it two when the former Tottenham Hotspur man had the ball in the back of the net, but the flag was raised after Deeney was penalised for his push on Charlie Taylor.
Watford see it out
After taking the lead in a fortunate manner the Hornets looked to wrap up the match as the match entered its latter stages.
Deeney had a good opportunity to double their lead with a little over 20 minutes to play when he got around Wootton but couldn’t get his shot off before eventually putting a ball into the far post, Leeds clearing the danger.
The last real chance came 11 minutes from the end when Holebas did very well to get to the edge of the area before pulling the trigger, but it was straight down the throat of Silvestri.
Leeds final opportunity to grab a replay came a minute from time when Daskoura received the ball but striker blazed it over and sealed Watford's passage into the last eight.