Livermore netted just before the end of normal time after Abel Hernandez's goal earlier in the second half had brought them level.
The Tigers had been out of sorts in a first half which saw Nick Blackman give the visitors the lead after Allan McGregor had pulled off two saves, before Hernandez and Livermore secured the win.
McGregor produced an early stop from Reading's Daniel Williams who had been presented with the chance following a slip by Hull defender Moses Odubajo.
McGregor then turned a shot around the other post from Blackman.
Hull were passing the ball well in midfield but their good build-up play was wasted by several final passes in a first half which saw them fail to threaten the Reading goal in the first 25 minutes.
Reading took the lead their dominance deserved after 28 minutes as Blackman fired left-footed into the roof of the goal after the Royals had worked the ball well across the Hull area to leave the over-lapping striker unmarked.
Five minutes later Sam Clucas' free-kick was headed down by Harry Maguire and fellow central defender Curtis Davies poked the ball towards goal only to see his effort blocked.
At least the home side had registered their first effort on target in what had proved to be a lacklustre performance so far.
Blackman fired in a free-kick which hit the top of McGregor's crossbar just before half-time.
Hull did not take long after the break to have their second shot on target as two minutes in Odubajo was played into the area by Hernandez but Jonathan Bond blocked his shot from close range.
Livermore shot from the edge of the area but it was straight at Bond and then Hernandez fired wide as the Tigers looked for a way back into the game.
Hull drew level after 61 minutes when Hernandez was on hand to fire home from a matter of yards after Bond had stopped Ahmed Elmohamady's shot with his legs.
Anton Ferdinand had blocked a Clucas effort but the winger slid the loose ball across goal to Elmohamady and when his shot was blocked Hernandez fired home for his eighth Championship goal of the season.
Chuba Akpom shot just wide as the home side sought another goal their second half play merited.
Akpom was then denied by a fine block by Bond after he had exchanged passes with fellow striker Hernandez.
Elmohamady headed into the side-netting from a cross by substitute Shaun Maloney as Hull's search for a winning goal went on.
The goal Hull had looked for finally came when Livermore shot in from close range after a goalmouth scramble.