Snodgrass put his team in control with the opening goal at Ashton Gate after 17 minutes before teeing up Max Gradel to seal victory in the second period.
The performance capped off a memorable week for the United schemer, who made his Scotland debut in midweek after coming off the bench against Northern Ireland.
Grayson led the chorus of praise for the 23-year-old as his team kept up their promotion push with another accomplished display.
The Leeds chief said: “It has been a huge week for Robert, with his first game for Scotland and now this excellent performance.
“He is a lad who deserves everything that comes his way because he works so hard at his game.
“In the 92nd minute today he was still closing down Bristol City’s full-back and playing his part for the team.
“Max Gradel is a similar type of player on our other wing and he also took his goal well and played a major role in our victory.”
Grayson refused to get carried away after his side put four points between themselves and seventh spot.
More tellingly for Leeds is the fact they are just half that amount shy of the top two, although the manager would not be drawn on an automatic promotion tilt just yet.
He added: “Bristol City had some chances and we had to make some last-ditch challenges so it was great credit to the lads that they kept a clean sheet.
“A lot of clubs still fancy their chances of automatic promotion in the Championship.
“We don’t look at other results too closely, but will go on doing our jobs and see where that takes us.”
Bristol City counterpart Keith Millen admitted his side were outplayed by Leeds.
City did have their chances but were punished as Snodgrass and Gradel combined to keep the Robins in the lower reaches.
Millen said: “We were second best.
“We deserved to score at least one goal, but you cannot defend like we did for the two Leeds strikes and expect to get anything from a game.
“There was a big build-up to the match for a number of reasons.
“We had a near-capacity crowd, which makes our performance doubly disappointing.”
Bristol City: James, Carey (Adomah 56), Caulker, Fontaine, McAllister, Elliott, Cisse, Woolford, Clarkson, Keogh (Maynard 67), Pitman. Unused substitutes: Gerken, Stewart, Johnson, Sproule, Nyatanga.
Leeds United: Schmeichel, Lichaj, O’Brien, Bruce, McCartney, Snodgrass, Kilkenny (Somma 78), Johnson, Gradel, Howson, Becchio (McCormack 85). Unused substitutes: Higgs, Paynter, Sam, Parker, Bromby.
Referee: J Linington (Isle of Wight).
Man of the match: Robert Snodgrass.