Tymon scored his first after 24 minutes, heading home a Michael Mackay cross and then after the hour when he converted a penalty after being brought down by Alan Martin.
Elliott had equalised just before the break when his volley from the edge of the box squeezed through the legs of Jan Budtz in the Hartlepool goal.
Leeds included 17-year-old trialist Shane O'Connor in their youthful starting line-up, with the Liverpool midfielder hoping to impress Gary McAllister.
A youthful Leeds side certainly started the brighter, with Pools failing to deal with the second ball from flick-ons by powerful 17-year-old striker Elliott.
United central midfielders Ollie Hotchkiss and Fabian Delph also impressed when given time on the ball and they looked to use the electric pace of right winger Seb Sorsa at every opportunity.
Sorsa sent Nathan Cartman away down the channel in the ninth minute, but his attempted chip over Budtz landed on the roof of the net.
Budtz was forced into action a minute later when Elliott got a yard ahead of Joe Tait and fired in a shot which the Pools keeper parried into the side netting.
Pools' best moments in the opening stages all stemmed through Jonny Rowell .Rowell constantly found space and was not afraid to put a boot in when necessary.
The central midfielder's skill enabled Pools to take the lead when he controlled a ball that he flipped over his own head and lifted an inswinging cross to the far post which Tymon met with a diving header into the far corner.
Pools almost handed Leeds an equaliser when Budtz came out of his box, tried to dribble round Sorsa and lost the ball.
Sorsa's attempt at goal was cleared off the line by Godwin Antwi. Budtz's blushes were not spared four minutes from half-time though as Leeds grabbed a deserved equaliser.
Elliott's half volley from the edge of the area should have been meat and drink, but Budtz didn't get behind it and the ball bounced past him into the back of the net.
Pools could have re-gained the lead six minutes after the interval when Rowell's free-kick from 25-yards was too good for Leeds keeper Martin.
Mackay followed up and got a touch to the ball, but it was cleared off the line. There was no denying Pools moments later when Craig Ruddy's through-ball sent Tymon away and he was clipped by the on rushing Martin.
Tymon stepped up himself and calmly slotted the spot-kick down the middle for what proved to be the winner.
Leeds: Martin, Gardner, Darville, Hotchkiss, Wilkinson, Milne, Sorsa, Delph, Elliott, Cartman, O'Connor (Ovington 62).
Subs: Edwards, Fisher, Falkingham, Connor.
Thanks to Chris Young of the Hartlepool Mail for the above report