Alejandro Garnacho gave Manchester United a lightning start after a shocking error from opposition goalkeeper Nikita Haikin.
But Hakon Evjen levelled for the Norwegians and Philip Zinckernagel surged down the right wing to give them a shock lead.
Hojlund pulled United level with a superb finish just before half-time and poked them ahead early in the second period.
Former Sporting Lisbon boss Amorim, who has said his methods will take time to implement, made six changes from the side that laboured to a 1-1 draw against Ipswich in his first game in charge on Sunday.
Tyrell Malacia returned to the United line-up for the first time since the final game of the 2022/23 season.
Bodo/Glimt also made multiple changes with one eye on domestic matters -- they are top of the Norwegian league table by a point with one match to come.
Old Trafford welcome
Amorim waved to the fans and was besieged by photographers as he took his seat at an expectant Old Trafford for the first time.
And within seconds he was celebrating a goal as United scored even earlier than they had at Ipswich -- within the first minute.
There appeared to be little danger as the ball was played back by Jostein Gundersen to Haikin but the goalkeeper let the ball run and Hojlund was on him in a flash.
Haikin scrambled to rescue the situation but the ball fell to Garnacho, who poked home for his eighth goal of the season.
The home side were on the front foot, with Mason Mount and Bruno Fernandes both forcing saves.
But Bodo/Glimt were playing some neat football and equalised following a fine move when Sondre Brunstad Fet found Evjen, whose rising finish gave Andre Onana no chance in the 19th minute.
Just four minutes later the bouncing hordes of yellow-shirted away fans were celebrating again, with United's fragile defence again exposed.
Patrick Berg freed Zinckernagel and the forward surged onto the ball, outmuscling and outpacing Malacia before finishing through Onana's legs.
United huffed and puffed as they tried to get back into the game and levelled when Noussair Mazraoui crossed for Hojlund, who killed the ball with his left foot and drove home with his right.
Amorim punched the air on the sidelines.
The Portuguese manager replaced the rusty Malacia at half-time with Diogo Dalot, switching from his favourite back three to a back four.
Mount hit the top of the crossbar after good work on the left by Garnacho.
And United were back in the lead in the 50th minute when Mount's clever flick found Fernandes, who crossed for Hojlund to tap in.
Amorim made more substitutions and another tactical tweak to a back three again.
United had more control than in the first half but were wasteful and faced a heart-stopping moment when Onana raced out of his area and just about won the ball as Bodo/Glimt threatened.
Onana was called into action in stoppage time to push out a Berg free-kick as United just about got over the line.
United sit 12th in the standings, with two wins and three draws from their first five games in the new-format league phase.
© 2024 AFP