Calvin Stengs gave Feyenoord the lead in first-half stoppage-time when his 30-yard free-kick evaded a poor attempt at a defensive wall.
Joe Hart saved a penalty after Lagerbielke received a second yellow card and substitute Holm soon got a straight red card in the 68th minute.
Alireza Jahanbakhsh got the home side’s second goal.
Centre-back Lagerbielke was one of two summer signings in the Celtic starting line-up with Nat Phillips ruled out by an ankle injury. The other, winger Luis Palma, made his first start for the club.
There was a lively atmosphere inside De Kuip for the first meeting between the teams since Feyenoord’s 1970 European Cup final triumph and the first Champions League game in the stadium for six seasons.
Celtic took the edge off the atmosphere with a positive start and Reo Hatate got the first shot on target after the Hoops won the ball high up the park, but his effort lacked power.
Celtic continued to look comfortable and could have made more of their possession had Hatate and Daizen Maeda been sharper on the ball at times.
The hosts rarely threatened to get in behind in the opening half hour and on the one occasion they did Alistair Johnston came across to cover.
Hatate’s pass got Matt O’Riley in behind but his ball towards Kyogo Furuhashi was cut out and Hart made his first save in the 34th minute when he pushed over Yankubah Minteh’s long-range strike.
Maeda broke away and saw a dangerous effort pushed wide despite being fouled in the act of shooting.
The Japan international was soon culpable for a counter-attack the other way after stopping in anticipation of a throw-in as Johnston kept the ball in, but Hart saved from Quiulindschy Hartman.
Celtic looked set to go into the dressing room deservedly level before O’Riley gave away a free-kick. Celtic’s wall failed to provide much protection especially as Furuhashi inadvertently got out of the way of the ball as he jumped. Stengs’ shot curled away from Hart with the goalkeeper unable to keep it out despite getting his hands on it.
Palma cut inside and saw his shot beaten away after the interval before Holm came on along with Yang Hyun-jun in the 58th minute.
Celtic were soon down to 10 men though. Lagerbielke had been booked in the first half for bringing down an opponent after his own poor control, and the Swede caught Igor Paixao in the throat with his arm as he tried to jockey the forward in the box.
Hart saved Paixao’s penalty as he dived to his right.
Brendan Rodgers brought on midfielder Tomoki Iwata for Furuhashi to play alongside Liam Scales in central defence.
Celtic’s task got even harder when Holm lunged in for a challenge and caught Mats Wieffer high up his shin. The visitors were rightly frustrated that Bosnian referee Irfan Peljto had failed to award a clear foul against Scales seconds earlier but there was little doubt about the red card decision that followed.
Stengs set up Lutsharel Geertruida to score from the resulting free-kick but the goal was ruled out for offside following a VAR check.
The extra men counted in the 76th minute though when Jahanbakhsh fired high into the net from 12 yards and the closing stages were about damage limitation for Celtic.