For Manchester City this was the night they learned what a tough, uncompromising competition the Champions League can be and, in the process, reminded us how far, for all the money, ambition and forward momentum, they still may be from being a club with realistic aspirations of glories in Europe.

With only two games played, the damage is still reparable but their chances of progressing from Group A are already beginning to look flimsy, and the controversies surrounding this match hardly speak of a club who possess the togetherness and unity that every successful team enjoys. On the contrary Carlos Tevez's behaviour strongly suggests a club with too many fractures – one that will, in the end, be better off without the Argentinian.

Tevez deserves all the criticism he attracts for the remarkable act of self-absorbed petulance that saw him refuse to come on as a second-half substitute, apparently because he was angry that he had not been asked to enter the play earlier. This was open indiscipline at its worst and, whatever Tevez says in reply, the man who lifted the FA Cup in May has somehow manoeuvred himself into a position where he is destined to be remembered for all the wrong reasons at Eastlands.

But this told only part of the story and Edin Dzeko's part in an ignominious night cannot be overlooked either. The striker's reaction to being substituted was reminiscent of a teenage brat, from the sarcastic thumbs-up to his manager to the stream of invective he directed towards Mancini before throwing his tracksuit top on to the floor and storming off down the tunnel. He, like Tevez, will have his own version of events – be it an apology or a defence – but whatever is said will not alter the fact that it was unacceptable behaviour.

Mancini looked ashen-faced, trembling with anger, as he reflected on the traumas of a night when City paid a heavy price for losing their way at the end of the first half when Mario Gomez scored the goals that leave them with only a solitary point to show from their first two matches.

The strange thing is that Mancini's men began the game well. They looked comfortable on the ball, full of running and, for the first half an hour, pinned their opponents back.

But the Champions League is an unforgiving place to make mistakes and City made lots. All the control they had exerted in the opening exchanges was replaced by a recklessness that left Mancini dragging his fingers down his face in frustration. Their marking, in particular, was poor in the extreme, both goals stemming from rebounds after some splendid goalkeeping from Joe Hart. It was perhaps typical of City's defensive performance that a Bayern player should be first to the loose ball on every occasion.

The first goal was particularly tough on Hart after he had got his hand to Franck Ribéry's low, diagonal 20-yard effort. Hart was up in a flash to keep out Thomas Müller's follow-up with an even better save from point-blank range. He deserved better than the ball dropping to another Bayern player – and this time Gomez gave him no chance.

City's big night was suddenly turning into an ordeal.Hart, again, was blameless again as Toni Kroos swung a free-kick into the penalty area and Daniel van Buyten flashed a header goalwards. Hart kept the ball out brilliantly at his near post but Gomez followed up to turn in another close-range goal. City, a team that pride themselves on defensive parsimony, looked vulnerable to the point of naivety.

To give them their due, they were playing the Bundesliga's top club, a genuine European superpower embarking on their 151st match in the Champions League compared to two for their opponents. City had a lot of the ball but spent a lot of time on the edges of the Bayern penalty area without creating any clear scoring opportunities. Dzeko will probably reflect he should have made more of an early chance and there were a couple of moments when Jérôme Boateng, facing his former club, threatened to give way a penalty – but little else. Maybe Dzeko had a point that it was peculiar for Mancini, with the score at 2-0, to replace him.

Mancini, however, made it clear he felt his striker had played woefully and it was also the Bosnian's clumsy challenge that led to the free-kick from which Bayern scored the second. After that, Mancini responded to Tevez's fit of pique by bringing on James Milner for Samir Nasri and, finally, Aleksandar Kolarov for Gareth Barry. The substitutions were almost as confused as City's defending during those moments of a night when the club, frankly, embarrassed themselves.