With extra time looming, Schalk made the most of a bungled Fontaine clearance to tap in from close range.
Ross County, from the Highlands town of Dingwall with just over 5,000 inhabitants, won promotion to the Scottish Premier League in 2012 and had eliminated Celtic in the semi-finals.
Until the 1994-95 season, County played in the Highland Football League.
In 2010 they reached the Scottish Cup final, the country's more prestigious cup competition, losing 3-0 to Dundee United.
Sunday's victory means both Scottish knockout trophies are in the hands of Highland clubs, with Inverness Caledonian Thistle the Scottish Cup holders
(Reporting by Alan Baldwin, editing by Ken Ferris)
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