United twice came from behind to earn a 2-2 fourth-round draw at Tannadice.
Christian Doidge fired Hibs ahead early on but Lawrence Shankland converted United's first chance in first-half stoppage time.
Martin Boyle struck from long range just after the break but Appere matched his effort in the 74th minute.
The teams will meet again at Easter Road a week on Tuesday.
Boyle ran in from the right and his reverse pass set up Doidge to finish from 14 yards.
The hosts were soon badly exposed on the counter-attack and Boyle set up Melker Hallberg but United goalkeeper Benjamin Siegrist got down well to stop the Swede's shot.
United – who have taken 34 points from their last 12 Ladbrokes Championship games – began to force some territorial pressure without troubling Ofir Marciano in the Hibs goal and Kamberi saw a shot saved at the near post.
The equaliser came out of the blue almost three minutes into first-half stoppage time.
Ian Harkes got in possession and, with Hibernian's centre-backs being attracted to the ball, he played a pass for Shankland to run in behind. The Scotland striker held off Lewis Stevenson, waited for Marciano to commit himself, and slotted home his 26th goal of the season.
Hibs took less than two minutes of the second half to regain the lead. Kamberi made progress down the left before Scott Allan laid the ball back to Boyle 25 yards out. The winger took a touch and drilled the ball into the corner of the net.
The visitors had brought on 2016 Scottish Cup hero David Gray at half-time for the injured Jason Naismith as the right-back made his first appearance since suffering a knee injury in August.
Kamberi forced a save from long range and Hibs looked well in control, only for United to level again out of the blue in the 74th minute.
Substitute Appere sold Steven Whittaker a dummy on the left flank, cut inside and curled in off the far post from 25 yards.
The 20-year-old soon threatened to repeat the feat after good running and skill but was several yards wide with a similar curling effort.
Hibs twice came close to a stoppage-time winner but Siegrist denied Doidge on both occasions.