The result lifts County into 10th place in the Premiership, two points above St Mirren who now occupy the relegation play-off place above Hearts.
After a quiet opening, Filip Kiss had the first serious attempt of the evening, his volley flying wide of Alan Mannus' goal.
County went ahead through Dutch midfielder De Leeuw (30) who worked some space before crashing a low shot into the bottom corner.
The home side's best chance of an equaliser before the break fell to Dave Mackay - but the home captain elected to head across the face rather than try to test Mark Brown in the County goal.
David Wotherspoon's dangerous cross produced some anxiety in the County box before Stevie May - hitherto subdued - tested Brown with a powerful overhead kick.
County continued to keep the home side at arm's length until Mackay's deflected shot again brought the best out of Brown.
A linesman's flag ensured Steven Anderson's header deep into injury-time counted for nothing as County held out to claim only their second away win of the season - and three potentially crucial points.