Former Livingston midfielder Snodgrass fired the Whites in front just before the break with his first of the season only for Marvin Sordell to hit back for Watford at the death.
However, Snodgrass repeated the trick as the game went to added time in almost a carbon copy of his first effort as Leeds maintained their unbeaten start to the season.
Watford famously downed Leeds during the 2006 Championship play-off final and almost got off to the perfect start when striker Will Hoskins headed just wide in the opening minutes.
However, it was all Leeds from there with sought after striker Jermaine Beckford and Snodgrass both going close on numerous occasions.
But the hosts didn't have to wait long for the deadlock to be broken with Snodgrass slotting home from deep inside the Hornets penalty area in the 38th minute.
Leeds flew out of the blocks after the restart and were only prevented from going two up thanks to Watford keeper Richard Lee.
He first saved comfortably from Beckford in the 50th minute before stopping Snodgrass in his tracks five minutes later.
However, the visitors slowly crept their way back into contention and, when Manchester United loanee Tom Cleverley saw his shot cannon of the post moments later, a comeback was on the cards.
Watford had to wait until 87th minute to get their reward but it was worth the wait with youngster Sordell firing home from close range.
But it wasn't to be for the Hornets with Snodgrass handing Leeds victory barely eight minutes into extra-time.