Stuttgart narrowly missed out on automatic qualification for the round of 16, but this clinical away performance has put them firmly in control of the tie.
"Our players knew what was at stake today," Hoeness said after the match. "They delivered a very focused performance. We had phases where we were under pressure and had to defend as a team.
"We played with great control for long stretches. We showed efficiency. In the end, we scored a very important goal to make the result even more decisive. Now we want to seal the deal and advance to the next round. We won't make the mistake of thinking it's all over now. We have to be fully ready again in a week."
Stuttgart struck first in the 15th minute through Bilal El Khannouss, who capitalised on a poor clearance from Kasper Schmeichel after a flick-on by Deniz Undav. Celtic responded six minutes later when Benjamin Nygren punished a mix-up between Atakan Karazor and Alexander Nubel to make it 1-1.
The visitors quickly regained control. El Khannouss grabbed his second in the 28th minute, rising unmarked to head home from close range after a deflected cross by Angelo Stiller.
Jamie Leweling extended the lead in the 57th minute with a powerful low strike from around 22 yards that slipped past Schmeichel. Tiago Tomas then wrapped up the win deep into stoppage time, surging into the box and lofting his finish over the goalkeeper to make it 4-1.
Stuttgart now turn their attention back to Bundesliga action against bottom-placed Heidenheim on Sunday before hosting Celtic in the return leg next Thursday. Should Hoeness' side complete the job, they will face either Porto or Braga in the round of 16, with the draw scheduled for the day after the play-off round concludes.
FC Barcelona presidential candidate Xavier Vilajoana has revealed that exploratory talks have already been held over a potential move for Harry Kane, insisting the Bayern Munich forward would be a "perfect fit" for the Catalan giants.
Michael Olise will step onto the pitch this weekend for Bayern Munich with more than just three points on his mind.
Borussia Dortmund head to RB Leipzig on Saturday knowing the stakes could hardly be higher in the Bundesliga title race.
Bayern Munich coach Vincent Kompany says his focus is only on the Bundesliga match against Eintracht Frankfurt at Allianz Arena on Saturday rather than upcoming derby.
