The visitors snatched the lead against the run of play in the 24th minute. Simon Davies darted down the left before picking out skipper McBride, who slid the ball home from six yards.
The Cottagers went desperately close to adding to their lead. McBride, Brede Hangeland and Jimmy Bullard all struck the Reading crossbar in the second half but the points were secured in the final minute.
Substitute Nevland was put through by Davies and made no mistake, sliding past Marcus Hahnemann to score his first goal since his January move to Craven Cottage.
Their first away win since September 2006 moves Fulham to within four points of safety, while Reading now sit only three points and two places above the drop zone.