The Switzerland winger completed a comeback success that was richly deserved, with his powerful 18-yard shot adding to Roger Assale's earlier leveller.
Alfredo Morelos impressively lashed Rangers ahead shortly before the break, the Colombian's class apparent throughout.
Yet the Glasgow side were second best for long periods in the Group G contest, played on the artificial pitch of Stade de Suisse.
Young Boys had the better early chances, with Jean-Pierre Nsame guilty of a glaring miss in the 17th minute when Fassnacht's cross was flicked into his path, only for the striker to make a muddle of a near-sitter.
Cameroonian forward Assale went closer with a deflected strike from 25 yards that arrowed barely a foot away from the top-right corner.
It was against the run of play when Rangers forged ahead in the 44th minute. Morelos rolled centre-back Cedric Zesiger and powered through before cleverly disguising his shot and firing into the left corner when goalkeeper David von Ballmoos anticipated the ball going the other way.
The Young Boys leveller came in the 50th minute, Assale prodding the ball in from close range after Ulisses Garcia drove over a powerful cross from the left.
Both sides had penalty claims waved away: Rangers after Morelos looked to be caught when chasing down a dangerous cross, and Young Boys when the ball struck the hand of Rangers defender Connor Goldson.
Gianluca Gaudino shot over for Young Boys and Nsame headed wide as the Swiss side pushed for a winning goal, having lost their opening game at Porto when Rangers beat Feyenoord.
Friday's (AEST) results, with Feyenoord beating Porto, mean all four sides are on three points from two games.
It needed a sound reaction save from Van Ballmoos to turn away a deflected Morelos strike in the 90th minute, as Rangers almost pinched the spoils.
But Young Boys would have the final say, Fassnacht cracking a left-footed shot through the weary defences of McGregor to leave Rangers, and their manager Steven Gerrard, dismayed.