City dominated for long periods and despite David Connolly twice hitting the woodwork, the visitors held firm.
Having survived normal time with their goal intact, Watford saved their best for the very last when, in the fourth minute of stoppage time, keeper Richard Lee hoisted a long ball forward and Bruce Dyer nodded on for Webber to keep his composure and score with a close-range drive.
Leicester almost got off to a flying start as early as the first minute when Danny Tiatto and James Scowcroft combined to set up Gareth Williams, whose 12-yard shot in-front of goal ricocheted to safety off the keeper's shins. It was to prove a costly miss.
The Foxes continued to pin back Watford with Sean Dyche and his fellow defenders working hard to keep the home side at bay.
Watford almost surprised City after 22 minutes when Webber got away from Matt Heath and forced Ian Walker into a diving save.
The visitors thought they had scored from the resultant corner when Brynjar Gunnarsson headed home, only for the referee to blow for a foul on the keeper.
Connolly then squandered a great chance to put the Foxes one up when he blazed over with the goal at his mercy.
The Republic of Ireland international was again in the thick of the action when his downward header from a Nikos Dabizas cross appeared a certain goal, until Lee intervened with an acrobatic one-handed save.
Watford made a bright start to the second half with Walker having to save with his chest after Webber set up Neal Ardley for a shot from 12 yards out.
Dyche then came to Watford's rescue when he cut out a dangerous Joey Gudjonsson cross with Connolly ready to pounce.
City almost broke the deadlock after 75 minutes when Williams hooked back in a half-cleared corner and Connolly reacted quickest to crash a close-range shot against a post.
Four minutes later, Connolly was again unlucky when he drove a Trevor Benjamin knockdown against the other post.
Lee rescued Watford three minutes from the end with a finger-tip save to keep out Danny Tiatto's 25-yard drive and then Connolly fired a 22-yard free-kick over the bar.
That was City's last chance and there was no way back for them when Webber scored his dramatic winner with just seconds left.