Despite waves of second half pressure and excellent individual performances from Dempsey Arlott-John and Sidnei Tavares, Ryan Cassidy's 100th-minute strike sealed safe passage for the Hornets.
Cassidy gave the visitors the lead inside 12 minutes from the penalty spot and, while a Harry Hudson own goal took the tie into extra-time, it's the Hertfordshire side who reached the semi-finals.
However, despite carving out the first opportunity, as Arlott-John’s header was saved by Dante Baptiste, they were behind on 12 minutes to a well-taken spot-kick.
Justen Kranthove was adjudged by the referee to have unlawfully felled Cassidy in the 18-yard box, and it was Watford’s No.9 who stepped up to drill it into the bottom corner.
Leicester City's new manager, Brendan Rodgers, was among the crowd.
Leicester’s youngsters, tough, responded well to the early setback with Arlott-John and Pennant both forcing Baptiste into fingertip saves from either side of the area.
At the other end, meanwhile, Edward Elewa-Ikpakwu needed to make a perfect challenge when Sonny Blu Lo-Everton was played in – and the City defender delivered just that.
The sides went in at the break with Watford holding a slender advantage, but it was City who emerged with the most attacking endeavour as Arlott-John twisted and turned in the box.
His low drive from a tight angle worried Baptiste enough to prompt the young Hornets shot-stopper to palm it away, instead of catching it, and was a sign of the home side’s intent.
Most of Leicester’s attacking play came through Arlott-John and the young striker took matters into his own hands just before the hour-mark, running at the visitors’ defence.
He galloped down the left wing, beating George Langston, and fired towards goal. While Baptiste was able to save, it clattered off the crossbar before going out for a corner.
Dempsey Arlott-John was particularly impressive for City.
With 15 minutes to go, Tavares slid through an inch-perfect pass for Arlott-John. Hornets 'keeper Baptiste, however, was there once again to make a low save.
Arlott-John was undoubtedly among City's most impressive players on the night and he was frustrated again by the visiting shot-stopper when his curler from 25 yards was tipped over.
Their refusal to accept defeat in normal time was finally rewarded with four minutes to play as Elewa-Ikpakwu raced down the right flank, crossed, and forced Hudson to send it into his own net.
That took the tie into extra-time and, within seconds, Arlott-John created another chance, prompting a last-ditch block from Hudson, before Baptiste saved from substitute Harrison Myring.
Drama followed in the opposite penalty area, too, as Lo-Everton cut inside his man and seemed certain to score - until Lukáš Hušek slid in to clear it off the line.
But Watford did restore their lead 10 minutes into extra-time as Emmanuel Adebiyi slotted a smart pass into Cassidy, who made no mistake.
City's visitors were defensively brilliant in the latter stages to see out the game and, despite a valiant effort from their hosts, Watford will face either Bury of Liverpool in the last-four.
City: Bosworth, Elewa-Ikpakwu, Thomas (c), James (Fitzhugh 64’), Kranthove, Hušek,, Pennant, Ewing (J. Gyamfi 110'), Russ (Myring 83'), Tavares, Arlott-John
Subs not used: Stolarczyk, Sams
Goal: Hudson OG 86'
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