Roy Keane's Black Cats appeared to have salvaged a point as Michael Chopra's late penalty cancelled out Ivar Ingimarsson's opener for the home side.
But Hunt's shot two minutes into stoppage time was ruled to have crossed the line before Craig Gordon's outstretched arm had pushed the ball out.
It was tough on Sunderland, but Reading would have kicked themselves if they had failed to win after creating a string of chances over the 90 minutes.
Hunt had come closest to opening the scoring in a first half dominated by the home side, but in a remarkable preview of what was to follow, Gordon had scrambled his effort away with Hunt claiming the ball had crossed the line.
It took Reading nearly 70 minutes to make their dominance count, Ingimarsson bundling home from close range, after Gordon had parried Ibrahima Sonko's header from a James Harper free-kick.
But Kevin Doyle, who had missed a hat-trick of first-half chances, then squandered an opportunity to make the game safe when he fired across goal after Dave Kitson had teed him up.
The home side were made to pay for their profligacy eight minutes from time when Kenwyne Jones raced clear of the home defence and crumpled under a sliding challenge from Sonko.
There seemed to be no contact between the players but, crucially, no contact with the ball either and referee Steve Tanner pointed to the spot, giving Chopra the chance to fire past Marcus Hahnemann.
Jones then had the chance to snatch victory for the visitors when he sprinted clear of Ingimarsson, but saw his effort blocked by an alert Hahnemann.
But there was still time for a dramatic finale and Hunt provided it with seconds to spare, arriving unmarked at the far post to meet substitute Shane Long's cross.
The linesman's flag immediately went up to signal a goal, despite Gordon's desperate attempt to keep the ball out, sparking huge protests from the Sunderland players on the final whistle.
But Reading had been the better team in a frustrating game and ultimately warranted the three points, however controversial the circumstances.