Liverpool hit the goal frame twice, through Jordan Henderson and substitute Stewart Downing, but missed a chance to go above Newcastle into sixth place.
It was Liverpool’s first loss in nine matches in the league, denting hopes of making ground on its rivals for the Champions League positions.
Manager Kenny Dalglish will point to Spearing’s dismissal as the turning point in the game because his team was dominating at that point.
The youngster was chosen as Liverpool’s defensive shield in midfield with regular starter Lucas Leiva having been ruled out for the season last week with knee-ligament damage.
He took his enforcer role too far, though, leaving his team with an uncomfortable final 20 minutes to see out. Dempsey ensured the visitors weren’t able do that.
With the match featuring two of the five meanest defenses in the league, the shortage of goals came as no surprise although Liverpool had come away from Craven Cottage with a 5-2 victory last season.
Maxi Rodriguez, the scorer of a hat trick in that match, was left out as Liverpool manager Kenny Dalglish opted adventurously for a front three of Andy Carroll, Craig Bellamy and Luis Suarez. But it was only in the second half that the trio came to life.
Henderson curled a shot against the far post in the 28th minute, yet it was Fulham which created the better of the chances in an even first half with Dembele and Dempsey twice having shots saved by Reina.
Liverpool upped the intensity after the break to pin Fulham back, with goalkeeper Mark Schwarzer saving from Luis Enrique and Suarez having a goal disallowed—unfairly, according to replays—for offside in the 67th.
Tempers had already risen thanks to a forehead-to-forehead confrontation between Bellamy and Dempsey at the start of the second half, but it boiled over in the 72nd when Spearing was dismissed.
There appeared nothing wrong with his initial full-blooded tackle on Dembele but he followed through with his other foot on the Belgian’s leg, leading referee Kevin Friend to brandish the red card.
Dempsey hit the bar, then Downing crashed a shot against the post from outside the area, before Dempsey struck the winner.
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