The Scottish side almost enjoyed the dream start when Nicolas Kuhn bent in a brilliant shot with less than 30 seconds on the clock, but an offside flag against Adam Idah ruined the moment.
Harry Kane made it two when he lashed in a powerful volley from a far-post corner but Celtic fought back as Daizen Maeda converted a close-range header in the 79th minute.
The Bundesliga leaders will take a slim advantage back to Germany but did not have everything their own way as the hosts chased an equaliser in a tense finale at Celtic Park.
Brugge, who scraped the final spot in the play-offs by finishing in 24th place, had taken the lead through Ferran Jutgla on the quarter-hour before Mario Pasalic levelled heading into the break.
Bayern Munich defeated a battling Celtic 2-1 in Glasgow to edge the first leg of their Champions League knockout round play-off.
