Slovakia now trail Group C winners Spain by five points, after the European champions beat Luxembourg 4-0 to clinch their place at the finals, and are level with Ukraine on 19.
Slovakia can guarantee a spot in France with a win away to minnows Luxembourg in their last qualifier on Monday due to a better head-to-head record than Ukraine who host Spain.
The Slovaks are aiming to qualify for the first time as an independent nation but have stumbled of late with one point in three matches after a perfect start to the campaign.
Stanislav Dragun put Belarus ahead in the 34th minute with a header from a corner that proved to be the winner for the visitors, who were already out of contention for the finals.
Coach Jan Kozak's side dominated possession but squandered several openings and failed to equalise even after Belarus's Aleksandr Martynovich was sent off after 65 minutes.
Belarus goalkeeper Andrei Gorbunov made several fine saves, including stopping a Vladimir Weiss free kick heading for the top corner and batting away a near own goal near the end.
(Reporting by Jason Hovet; editing by Ken Ferris)
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