The Premier League club suffered the upset on Tuesday as Bodo/Glimt secured a first Champions League victory thanks to Kasper Hogh's quick-fire double and Jens Petter Hauge's stunning solo effort.
City managed a consolation through Rayan Cherki, but the second-half dismissal of 2024 Ballon d'Or winner Rodri sealed a miserable evening in the Norwegian town of Bodo.
In the aftermath, senior City players Bernardo Silva, Ruben Dias, Rodri and Erling Haaland issued a statement acknowledging the sacrifice of supporters who made the long journey north.
"Our supporters mean everything to us... it was a lot of travelling for the fans who supported us in the freezing cold throughout a difficult evening for us on the pitch," they said on Wednesday.
"Covering the cost of these tickets for the fans who travelled to Bodo is the least we can do."
They will reimburse the 374 fans a total of 9,357 pounds ($12,568) after each paid about 25 pounds per ticket.
"The City support has an incredible connection with the players on a matchday, and this gesture is yet another reminder of that relationship - it means a lot to us," said Kevin Parker, speaking on behalf of City's Official Supporters Club.
"We know that the players are disappointed with the defeat to Bodo, but with our next game at home on Saturday, there's a chance to get back to winning ways, and our fans will be in full voice, backing the team as they always do."
Pep Guardiola's side, who lost 2-0 in the Premier League at Manchester United on Saturday, host Wolverhampton Wanderers on Saturday.
City sit second in the Premier League with 43 points from 22 matches, seven points behind leaders Arsenal.