It took Young Boys just five minutes of the second half to level things up as Roger Assale fired in from close range.
And the hosts broke Rangers hearts deep into added time when Fassnacht fired in to snatch all three points.
All four teams in Group G now have three points apiece after Feyenoord beat Porto in the pool's other game.
The sides cancelled each other out in the early stages as both teams enjoyed plenty of possession but created few clear-cut chances.
Jean Pierre Nsame missed the game's first meaningful opportunity when he stabbed the ball wide from close range following some good build-up play from Fassnacht and Assale.
Morelos darted into the box and shot at goal in the 36th minute but his tame effort was easily gathered by home goalkeeper David Von Ballmoos.
The Colombian went through on goal again one minute before the break after turning a defender and latching onto Borna Barisic's brilliant ball, and this time he found the back of the net as he fired beyond home goalkeeper Von Ballmoos to open the scoring.
Less than a minute later, Sheyi Ojo flashed a dangerous ball across goal from the left side but no one in a Rangers shirt was there to turn it home.
Young Boys equalised five minutes into the second half when Assale poked in Ulisses Garcia's cross from a few yards out.
Morelos worked a good opening in the 53rd minute, unleashing a fierce drive from the right wing and forcing Von Ballmoos into a good save.
Rangers appealed for a penalty four minutes later when Morelos appeared to be brought down in the area but the referee was not interested.
Young Boys midfielder Gianluca Gaudino chanced his arm from range in the 63rd minute but Allan McGregor made a comfortable save.
Connor Goldson headed over from a corner in the 77th minute as Rangers continued to push for a second.
Morelos almost snatched maximum points for Rangers in the 90th minute but Von Ballmoos produced a sensational stop to keep his shot out.
And Young Boys compounded their misery when Fassnacht steered home late in injury time to win the game for the hosts.