Ulreich made a bizarre mistake in the second half, as he attempted to collect Corentin Tolisso's back pass, only to seemingly realise that he was unable to use his hands. The back-up to No.1 Manuel Neuer subsequently slipped, allowing Karim Benzema to score his second goal on the night.
Bayern fought back to 2-2, through on-loan James Rodriguez, having led at the Santiago Bernabeu, but they were unable to find a decisive third goal, and were subsequently eliminated.
Widely circulated pictures showed Ulreich on his own on the Bernabeu turf after the final whistle, clearly distraught, and he has now issued a public apology to Bayern's supporters.
"Words can't describe how disappointed I am for being knocked out," he wrote on Instagram.
"We desperately wanted to reach the final and we gave our best and then this needless mistake happens to me. I can't explain it.
"I'm sorry for my team and for you fans."
Bayern's official Instagram account, as well as many of his team-mates - including Jerome Boateng, Javi Martinez and James Rodriguez - were quick to comfort Ulreich on the social media platform, however.
Liverpool hold a 5-2 lead over the Serie A side from the first leg, but will be wary of their Italian opponents, who overturned a similar three-goal deficit to knock Barcelona out in the quarter-finals.
Zinedine Zidane's side are no stranger to European finals, with Zinedine Zidane reaching the final in each of the last three seasons.
Indeed, the former France playmaker has yet to be eliminated from the Champions League during his time at Real.