Leeds made just one change from the side that defeated Tranmere at the weekend.
Mike Grella replaced Luciano Becchio in attack against Championship Watford.
In a quiet start to the game Leeds had to wait until fifteen minutes in before having a proper effort on goal, Bradley Johnson tried his luck from distance but his shot went over the bar.
Watford quickly had their first good chance, Cowie`s shot was superbly saved by Higgs and put out for a corner.
Snodgrass had a couple of chances but failed to make the keeper work with either effort.
Hoskins then had a good chance for the visitors but shot straight at Higgs.
Robert Snodgrass finally broke the deadlock when he bent it into the far corner with his left foot.
Leeds went in at half time happily 1-0 in the lead.
The home side started the second half the better and both Jon Howson and Robert Snodgrass went close.
Watford were still in the game though and John Eustace went close to equalising.
Mike Grella was once again making an impression in the side and he set up Beckford but Lee in the Watford goal turned his effort over the bar.
Leeds who had looked comfortable in the lead fell for the sucker punch with three minutes remaining when substitute Marvin Sordell equalised for the visitors.
The full time whistle went with Leeds disappointed not to have killed the game off in normal time.
Leeds retook the lead in the first period of extra time when Snodgrass scored his second of the evening latching onto a flick on from Showunmi.
Doyle nearly put Leeds further ahead when his speculative effort was turned away for a corner.
Showunmi had a couple of chances to increase the lead but failed both times when in a good position.
Leeds almost got a third in the final seconds when they hit the post.
Leeds held on to book their place in the third round draw and continue their impressive start to the season.
Teams
Leeds
Higgs, Crowe, Marques, Kisnorbo, Hughes, Snodgrass, Howson, Doyle, Johnson, Beckford, Grella.
Subs:
Ankergren, Robinson, Becchio, Prutton, Kilkenny, Showunmi, Michalik.
Watford
Lee, Mariappa, Hodson, Doyley, Williamson, Cowie, Eustace, Jenkins, Cleverley, Hoskins, Graham
Subs:
Loach, Severin, Lansbury, Harley, Henderson, Bryan, Sordell
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