Armed with a commanding three-goal advantage from the first leg in Glasgow, Die Roten returned to home soil as heavy favourites to secure a last-16 berth.
However, Sebastian Hoeneß’s side were dealt a reality check inside 30 seconds, when Junior Adamu’s clever pass was clinically swept into the bottom corner by Luke McCowan to give Celtic a dream start.
Determined to find an immediate response, Stuttgart ramped up the pressure as the first half progressed, with Badredine Bouanani seeing a long-range strike parried away by Viljami Sinisalo before the Algerian international misjudged another good chance at the back post.
Having scored 3+ goals in six of their last eight matches (W6, D1, L1), the hosts continued to probe in the final third, but the Celtic rearguard held firm, as Tiago Tomás’ goalbound effort was blocked in the six-yard box.
Stuttgart carried their momentum into the early stages of the second period, with HT substitute Chris Führich setting up Bouanani, whose close-range strike was repelled by the challenge of Marcelo Saracchi.
With time heading into the final half-hour, Celtic knew the importance of scoring the next goal; however, it was Hoeneß’s side who remained the more threatening, as Sinisalo reacted smartly at his near post to deny Bouanani.
The hosts thought their pressure had finally paid dividends in the 75th minute when Deniz Undav rifled home from the edge of the box, but the goal was ruled out by VAR for a marginal offside against Ermedin Demirović in the build-up.
Undeterred, Stuttgart – with the relative comfort of a two-goal cushion – looked to control the final 10 minutes.
Die Roten created plenty of chances late on, but Sinisalo once again stood firm in the visitors’ goal, denying Ramon Hendriks and Führich in quick succession.
Ultimately, the damage was done in the first leg for Celtic, as despite securing a first away victory in Germany at the 17th attempt, Stuttgart saw out the closing stages to progress to the last 16 of the UEL.
Flashscore Man of the Match: Viljami Sinisalo (Celtic)