The Scottish playmaker, regaining top form after an injury that ruled him out of the first seven weeks of the season, laid on an opener for the in-form Jonny Howson before netting a wonderful solo second.
Fourth-placed Coventry hit back after the break through Lukas Jutkiewicz and, although Max Gradel's penalty was quickly cancelled out by Ben Turner's goal to set up a tense finish, Leeds held on to build on last weekend's 4-1 win at Scunthorpe.
Howson scored a hat-trick that day and although he was again impressive, Snodgrass stole the show with a fine exhibition of wing play that had the Coventry defenders bedazzled all afternoon.
Leeds' preparations were disrupted when goalkeeper Kasper Schmeichel withdrew from the side on Friday night to be at the birth of his son, while Coventry were unchanged following their win at Sheffield United last weekend.
Schmeichel's absence did not hinder Leeds in the early stages, though, and they took the lead four minutes in through Howson.
George McCartney's cross from the left evaded everyone and found its way to Snodgrass at the back post, allowing the Scot to stand the ball up for the incoming Howson who planted a header beyond Keiren Westwood.
Neil Kilkenny then drove just wide as Leeds impressed and Coventry were fortunate to see another dangerous cross from Snodgrass fizz across their goal.
Jutkiewicz tried to get Coventry going with a header and a shot that inched wide, while Paul Connolly did well to get his body in the way of a goal-bound drive from former Leeds loanee Gary McSheffrey.
Howson could have doused the home side's fire had he directed a second header off another Snodgrass centre on target when he found himself free in the box, and the winger felt he should have been awarded a 28th-minute penalty when he went down under Martin Cranie's challenge, but referee Keith Stroud waved away his protests.
Jutkiewicz passed up a golden opportunity 10 minutes before the break as he found himself one-on-one with Jason Brown, only to allow Connolly to close his space and force him to shoot wide.
Snodgrass had been the game's star performer, though, and he deservedly doubled his side's advantage after 40 minutes, cutting in from the right and curling home a stunning 20-yard strike that gave Westwood no chance.
Coventry did have chances to reduce the deficit before the break, but Jutkiewicz was denied by Neill Collins before Brown held a Richard Wood bullet on his line.
Former Everton striker Jutkiewicz had been his side's best outlet all afternoon and it was he that gave them a lifeline in the 52nd minute with his fourth of the season.
With McCartney off the field injured, McSheffrey whipped in a cross that Turner nodded back across goal for the waiting Jutkiewicz.
Leeds manager Simon Grayson was unhappy that McCartney had not been allowed back on to the field in time and he and his coaching staff jostled with the Coventry bench in the aftermath.
It was soon Coventry's turn to complain as a penalty was awarded for a Richard Keogh's foul on Gradel, but their claims were without justification and the Ivorian winger dusted himself down to slot home from the spot (61).
But Leeds were as fallible in defence as they were lively in attack, and Coventry were back to within a goal just three minutes later as Turner prodded home a loose ball in the visiting area.
Turner was quickly proving his worth at his own end just after that, getting back to dive in the way of Luciano Becchio's effort which was destined for the goal after he had broke free and rounded Westwood.
Coventry laid siege to the Leeds goal for the remaining 26 minutes but, apart from a Wood header that Brown saved, they were unable to find a way to goal.