The Scottish winger hit Leeds' extra-time winner after seeing his 38th-minute goal cancelled out four minutes from the end of normal time by Watford substitute Marvin Sordell.
Grayson said: "It was very pleasing to beat a Championship side and I was very pleased with the way we played.
"Watford started the game better than us but we got to grips with it, Snod got his goal and we showed great resilience.
"The game should have been dead and buried by the time their goal went in but the players kept going. Snod has nearly been there in the last few games. He reached very high standards last season and his goals tonight will give him a big boost."
Watford manager Malky Mackay revealed that winger Tommy Smith is in the process of talking to Portsmouth and Reading who have had bids accepted for him.
Mackay added: "I was disappointed we were beaten but it was an exciting cup tie. I was proud of my players in terms of their courage and the way we drove on to the end.
"I was delighted with Marvin Sordell who took his goal well and was a real handful. His link-up play was very good too. Leeds are on the crest of a wave and are particularly strong at Elland Road, but for long periods of the first half we played some good stuff and their goalkeeper (Shane Higgs) had to make five exceptional saves."
Leeds savoured partial revenge for their 3-0 Championship play-off final defeat by Watford three years ago. Nothing could completely compensate for missing out on a return to the Premiership in 2006, but the first meeting between the clubs since that play-off final at Cardiff saw League One side Leeds take the bragging rights as Snodgrass maintained his team's 100 per cent record.
For his first goal he took American Mike Grella's pass and sprinted into the penalty box to find the far corner from 15 yards.
Winger Tom Cleverley hit the outside of a post in the 58th minute for the Hornets but it needed instinctive saves by Richard Lee to prevent Snodgrass and Jermaine Beckford from increasing Leeds' advantage.
With normal time running out, Watford attacked through the centre of the Leeds defence and Sordell stabbed the ball past Higgs after his first attempt was blocked. It was the first goal Leeds had conceded from open play this season.
But Snodgrass would not be denied and eight minutes into extra-time he steered the ball past Lee from 12 yards after latching on to substitute Enoch Showunmi's long pass.
Lee stopped a vicious volley from Michael Doyle and twice dived bravely at Showunmi's feet to prevent further Leeds goals.