With extra-time looming, Jonny Howson played the ball out to Robinson whose shot from the right flew over keeper Arran Lee-Barrett to send Leeds into the fourth round for the first time since 2001.
Neil Kilkenny and Andrew Hughes were the only starting line-up survivors from Leeds' win at Carlisle, while in contrast Hartlepool were unchanged.
Leeds had scored nine goals in their previous two Carling Cup ties this season but they fell behind in the second minute.
Joel Porter helped Ritchie Humphreys' free-kick towards the far post where Leeds-born Andy Monkhouse climbed unopposed to head into the net.
The lead lasted just 12 minutes as Hartlepool defender Michael Nelson lost possession to Robert Snodgrass, who sprinted into the penalty box and tucked his shot just inside the left post.
Ironically, Snodgrass was wanted by Hartlepool before his move from Livingston in July.
Hartlepool had been warned in the fifth minute when Snodgrass linked cleverly with Ben Parker only for the left-back to fire wide of the far post when well placed.
However, the visitors regained the lead in the 33rd minute when Australia international Porter spotted Casper Ankergren off his line and scored from 25 yards.
Snodgrass left the field with a stomach upset with top-scorer Jermaine Beckford replacing him in the 36th minute.
Ankergren pulled off a desperate save from Ritchie Jones just before the break to prevent Leeds falling further behind, but they levelled again on 57 minutes when David Prutton's cross was lifted on to the bar by Beckford and Enoch Showunmi forced home the rebound.
Hartlepool lost Antony Sweeney who was carried off after a 69th minute clash of heads with Parker who was able to continue.
The game seemed destined for extra-time until Robinson's late winner.