Gordon Strachan's men headed into the match already out of the race for Brazil in 2014, massive underdogs to claim a positive result and a staggering 70 places behind Croatia in the world rankings.
But Snodgrass proved to be the difference between the two nations when he grabbed the only goal of the game to hand the Scots their first competitive victory since beating Liechtenstein in October 2011.
Leigh Griffiths - who impressed on loan at Hibernian last season - made his first start for Scotland after being tasked with leading the line.
In the middle of the park for the clash in the Maksimir Stadium were James McArthur, Barry Bannan, captain for the night James Morrison, Shaun Maloney and Snodgrass.
Allan McGregor was between the sticks, while Russell Martin and Grant Hanley were given the nod in the heart of defence, with Alan Hutton and Steven Whittaker in the full-back positions.
The pre-match entertainment included three men parachuting onto the pitch, carrying the flags of both nations and the match ball.
Once the game was under way, Scotland enjoyed a decent amount of possession in the opening spell.
The visitors had an early chance to put some pressure on their hosts with a free-kick 30 yards out, which was played short by Bannan to Morrison - who rifled well over.
Ivan Perisic then had a go from distance at the other end but he, too, was well off-target.
The breakthrough came after 26 minutes and the Croats were stunned when it was the Scots who surged into the lead.
Fine play by the impressive Maloney resulted in the ball breaking for Snodgrass, who showed good anticipation to pounce on the loose ball and stab home.
The home side should have levelled when Sammir collected a short backpass by Hanley but blazed over with the gaping net at his mercy to jeers from the crowd.
Skipper Darijo Srna was then denied a goal by a timely clearance from Martin just before half-time.
Croatia continued their search for the leveller after the restart and Ivan Rakitic controlled the ball well before unleashing a shot that dipped just over the crossbar.
Ivica Olic then demonstrated good movement but was also unable to keep his effort down and McGregor was untroubled in goal.
Scotland could have added to their slim advantage when Bannan raced into the box before sending a dangerous swerving shot just past the far upright.
Croatia still looked dangerous and Olic whipped a tempting ball across the face of goal for Mario Mandzukic but he just failed to connect at the back post.
The home side continued to pile on the pressure and McGregor came to the rescue when he blocked a powerful shot from Mateo Kovacic with his legs.
As the tie entered the final 10 minutes, the Scots provided a reminder that they still posed a threat in front of goal as well.
Craig Conway burst down the left flank before picking out fellow substitute Steven Naismith in front of goal but he connected awkwardly with the header.
The ball then broke for Snodgrass, who failed to capitalise, but Scotland were able to hold on for an impressive win thanks to his goal.