A Charlie Adam free-kick was deflected into own net by William Kvist Jorgensen for the opening goal before Christian Eriksen netted for the Danes.
But it was Snodgrass who had the final say against a nation who had prevented the Scots from scoring in their last five meetings.
The victory proved to be the perfect dress rehearsal for Craig Levein's men as they prepare to host both the Czech Republic and Lithuania in a crucial Euro 2012 qualifying double-header at the same venue next month.
Leeds winger Snodgrass earned his first start having previously made two substitute appearances and, before the match kicked off, Scott Brown received a silver medal to mark his 25th cap for the national team.
Kenny Miller was tasked with providing the goals as the lone striker in a 4-5-1 formation, while Nicolai Boilesen made his debut for the Danes.
A relentless downpour all day had no doubt contributed to a smaller than usual crowd of 17,582.
The home side tried to give those who had made the effort on a wet Wednesday night something to cheer by putting their visitors under pressure early on.
Steven Naismith slipped a neat pass through to James Morrison in the box but a well-timed tackle from Simon Kjaer cleared the danger before the West Brom player could pull the trigger.
Snodgrass then delivered a tempting cross from the right for Miller but the Cardiff striker's header was weak and comfortably gathered by goalkeeper Thomas Sorensen.
There was concern for both sides when Brown and Kjaer collided with each other while both going in for the tackle and, although they returned to the pitch after receiving treatment, the Celtic skipper was then forced out of the action after just 20 minutes.
Denmark had a decent chance to open the scoring when Eriksen burst towards the box before setting up the shot for Dennis Rommedahl but goalkeeper Allan McGregor pulled off an impressive save to keep the scoreline level.
Brown had been replaced by Don Cowie, who contributed to the opener when he won a free-kick which was whipped into the goalmouth by Adam and helped into the back of the net by a deflection from Kvist Jorgensen after 21 minutes.
The Danes were back on level terms nine minutes later thanks to a set-piece of their own after Michael Krohn-Dehli was fouled by Adam out on the left flank. Eriksen then sent a lovely swerving free-kick past the grasp of McGregor and into the top corner.
The Rangers number one was called into action again, this time to block one-handed from Krohn-Dehli's point-blank shot.
But it was the Scots who found the back of the net again two minutes before half-time when Miller supplied a long ball from the left and Snodgrass did well to send a diving header into the back of the net from a few yards out.
Denmark emerged after the restart determined to haul themselves back into the game and substitute Lasse Schone tried his luck with a powerful shot from distance but McGregor was up to the task and parried wide.
Both managers used the second half to make a number of substitutions, with Craig Mackail-Smith, Graham Dorrans, Barry Bannan, James Forrest and Grant Hanley all handed a taste of the action for Scotland.
With the clock ticking down, Eriksen tried to add his own tally for the night with a long-range effort that deflected past the upright and failed to trouble McGregor.
Nicklas Pedersen then hooked over from close range but the visitors failed to find the goal that would allow them to return to Denmark with the draw, before Bannan drew a decent save from the goalkeeper with his own ferocious shot in the final minutes.