Leeds' unbeaten run had looked increasingly precarious at stadium:mk until Dons winger Jason Puncheon was sent off just before half-time.
Leeds struggled to take advantage of their extra man but deep into injury-time Snodgrass headed Bradley Johnson's left wing cross to seal the win.
Puncheon came close to scoring before half-time with a 20-yard drive, which Higgs parried away. But moments later the Dons winger was sent off for after a challenge on Michael Doyle.
It could have been a double disaster for the Dons but Leeds top scorer Jermaine Beckford fired over the bar from six yards out.
Leigh Bromby then headed wide from six yards and Michalik saw his shot blocked by Dean Lewington. But after surviving for more than 45 minutes with 10 men, Snodgrass stole it for Leeds right at the death.