Second-half strikes from Adam Johnson, Mario Balotelli, Samir Nasri and Stefan Savic eased City's midweek European disappointment and heaped more woe upon Kean.
Fans called loudly for his dismissal after Nasri knocked in the third goal and a section of the crowd produced 'Kean Out' banners after a result that leaves Rovers second from bottom in the Premier League.
But Kean, asked if he intended to see the job through, replied: "100 per cent". He added: "If the fans want to vent [their frustration] I'd rather they vent it at me than the team.
"I told the lads I will battle on for my players. Even the ones I didn't bring to the club I consider my players. I'll take the criticism on my shoulders and take it away from them."
Kean believes the criticism is particularly unfair given his side's heroic win over Arsenal a fortnight ago, and the fact they were simply undone by City's awe-inspiring quality.
Kean added: "A couple of weeks ago it was very buoyant dressing room. Obviously we're disappointed with the manner of the goals because I felt in the first half our game plan was working well.
"A couple of weeks ago we played Arsenal and got a fantastic result and most of the fans went home very, very happy. We've lost today and the fans are not happy, nor are the players, nor am I.
"But I think we have to put it in perspective. For the first 55 minutes the shape of our team was good and we minimised the chances of Manchester City, and it has taken a very good goal.
"We reacted by feeling a little bit sorry for ourselves and conceded another goal. It is certainly not something I am enjoying but I can take it on the chin."
City boss Roberto Mancini declined to attend the post-match press conference and was replaced by first team coach David Platt, with his arrival preceded by a ban on questions about Carlos Tevez.
Platt said he was delighted with the manner of his side's performance after a difficult week.
Platt said: "We wanted to win a football match and that's what we've prepared for since we got in on Thursday. We put Bayern Munich to bed and concentrated all our preparations on getting a result.
"We're blessed that one of the hardest things we've got is picking an 11 to go into a match. We know we can make six changes like we have done today and you don't lose that quality of performance."
Platt admitted there was some concern over a groin injury sustained by star striker Sergio Aguero which caused him to limp off after half an hour.
Platt added: "I don't think it's as bad as we first thought. Having said that if we had another game in the week he wouldn't play. We will have a good look at it over the next 48 hours."
And Platt was full of praise for the influential Balotelli, who was preferred to start alongside Aguero with Bosnian Edin Dzeko the one to make way.
Platt added: "Mario has had a lot of column inches in the last 12 months or so and a lot of it has been negative. That's fine but I think he deserves the plaudits as well."