Steven Gerrard's team were hoping to make the most of their noisy home support before travelling to next week's closed-door clash at the BayArena.
It remains to be seen if that game will even go ahead as the coronavirus crisis continues to sweep Europe, but if it does, Gers will find themselves with an almighty mountain to climb.
George Edmundson's header raised hopes Rangers could complete another remarkable rescue mission but they were snuffed out by Leon Bailey's late clincher.
Bosz had called for the tournament to be put on hold after claiming it was being devalued by postponements and matches staged in empty stadiums – but it is unlikely he will be saying the same now after seeing his team take a massive step towards the quarter-finals.
Rangers were again cast as the European underdogs – a role they have certainly relished more than the burden of having to break down stubborn domestic opponents of late.
But it was immediately obvious this was a step up for the confidence-stricken Light Blues.
Leverkusen were just as comfortable on the ball as Braga were in the last round but, unlike the Portuguese side, they had no obvious flaws in defence.
Borna Barisic had to put in a brave block on Karim Bellarabi just nine minutes in before Glen Kamara copped a booking for a foul on Mitchell Weiser which rules him out of the Rhineland return.
With the impressive Kerem Demirbay pulling the strings, Die Werkself continued to probe looking for a chink in Rangers' armour.
But it was a cruel call that eventually gave Leverkusen their lead eight minutes before half-time.
Edmundson did his best to hide his arm as Charles Aranguiz whipped in a cross from the right but under FIFA's strict definition it was a handball.
Gerrard hoped the incident had been missed, but once the VAR officials called Polish referee Szymon Marciniak over to the touchline in front of the Main Stand to view the pitch-side monitor there was no hiding it, with Leverkusen skipper Havertz coolly hushing the boos as he rolled the ball down the middle as Allan McGregor sprang to his left.
Ibrox appealed for their own helping hand from the video officials moments later when a Goldson header was blocked by Jonathan Tah, but it was ruled to have hit the defender's back.
The Light Blues boss did introduce the Swiss striker for Joe Aribo just seven minutes after the restart as he looked for a spark.
It almost came too. James Tavernier rolled in Morelos but the striker drew a blank again as Lukas Hradecky palmed wide. It then took a vital Tah intervention to deny the striker after Kamberi did well to rob Edmond Tapsoba.
Leverkusen's threat remained, however.
Aranguiz thought he had scored on 66 minutes with a hook shot until Steve Davis headed off his own line. But his celebrations were only delayed by a matter of seconds as he smashed a loose Tavernier clearance past the motionless McGregor.
Gers had fought back from two-down to stun Braga, and hopes of another remarkable comeback were sparked with quarter of an hour left as Edmundson was left unmarked to nod home from a Tavernier corner.
It might have been on had Barisic's free-kick not crept just over the Germans' bar. Aranguiz came closer at the other end as his set-piece shook the woodwork above McGregor's head.
But Gers' dreams were crushed two minutes from time when Bailey tormented substitute Matt Polster one way then another before sweeping a sublime finish past McGregor.