Extra-time seemed certain until winger Robinson accepted Jonny Howson's pass and blasted an unstoppable shot past startled keeper Arran Lee-Barrett.
Leeds-born Andy Monkhouse headed Hartlepool in front inside two minutes. Richie Humphreys' free-kick from the left was shunted to the far post by Joel Porter and Monkhouse headed past keeper Casper Ankergren.
Leeds were gifted a soft 14th-minute equaliser when Livingston striker Robert Snodgrass won the ball, sprinted forward and shot inside the left post from 15 yards.
Soon after Australian international Joel Porter had rifled Hartlepool back in front on 33 minutes with a 25-yard drive which caught Ankergren off his line, Leeds' leading scorer Jermaine Beckford left the bench in place of Snodgrass who was suffering from a stomach upset.
Leeds were much livelier in the second half and they pulled level again in the 57th minute, Enoch Showunmi forcing the ball over the line after David Prutton's cross was turned against the bar by Beckford before Robinson provided his late winner.