With Kylian Mbappé injured, Rodrygo put Madrid ahead in the first half -- ending a 32-game goal drought -- before Nico O'Reilly leveled for City, and an Erling Haaland penalty put the visitors ahead.
Madrid and Alonso went into the game under pressure after a 2-0 home defeat to Celta Vigo on Sunday made it one win in five games in LaLiga, with ESPN reporting that club executives had met to discuss the coach's future.
"When you don't win at home [whistles] can happen," Alonso said in his postmatch news conference. "We felt the energy from the crowd, when the team needed it, pushing them on. At the end of the game, if there were whistles we can understand it and accept it. The demands are so high."
Madrid began the season well, winning 11 of their first 12 LaLiga games, before a Champions League loss at Liverpool last month sparked a slump that has seen the team draw with Rayo Vallecano, Elche and Girona before being beaten by Celta.
Alonso's management style and the performances he has secured from key players have been scrutinized, while Madrid have also faced an injury crisis that saw eight first-team players including Mbappé miss the City game.
"The team kept going, we tried until the end, and with all the difficulties we have with injuries, the players gave everything," Alonso said Wednesday. "Today I have nothing to criticize.
"The results in November and December haven't been good, or what we wanted. We're self-critical, but we have to keep working and believing it will pass."
Midfielder Jude Bellingham said the players were "100 percent" behind the coach.
"The manager's been great," Bellingham told TNT Sports. "I've personally got a great relationship with him, a lot of the lads do as well. After the first run of games where we drew a few, we had some great conversations internally. In the last couple of games, we've just let ourselves down again. But no one is downing tools, no one is complaining, moaning, thinking the season is over."
Goalkeeper Thibaut Courtois denied the City clash had been an "exam" for Alonso.
"We don't see it that way," Courtois told Movistar. "We want to win because we know the situation we're in. We've dropped too many points in LaLiga, and although we're doing well in the Champions League, we can't afford to fall out of the top eight. But I think today we showed that we're behind the manager."
Alonso praised goal scorer Rodrygo, who embraced the coach after scoring, and wouldn't say if Mbappé would be available for Madrid's weekend game at Alavés.
"We still don't know if he'll make it for Sunday," Alonso said. "But today he wasn't able to play."
