And he reiterated support for the Black Lives Matter movement will continue to "help and move forward together as one".
Just as the players from both clubs were taking a knee in support of Black Lives Matter, a plane appeared, bearing the message 'White Lives Matter Burnley'.
There has been universal condemnation for the stunt, with the club promising lifetime bans for those involved.
Burnley-born Rodriguez wasn't alone in feeling upset, and said the players were firmly behind skipper Ben Mee, who said he was "ashamed and embarrassed".
Mee was speaking for the dressing room, and Rodriguez said: "He was, we had the conversation straight after, we were upset about it, gutted about the situation.
"It doesn't reflect what us as lads, us as people, the club, the town, think and live by.
"I'm proud of what he said, as a captain, as a person, it affected all of us.
"You always look to your captain for guidance, and he set the tone.
"But I feel it came from the lads and the staff as well in the changing room, that it wasn't acceptable, and we were gutted.
"The result was poor and not what we wanted - Man City can do that to any team, they are world class. But the changing room was quiet and everyone was sad about the situation.
"We are fully behind Black Lives Matter and it was totally against what we believe.
"It was the first thing we spoke about at half time. It is not acceptable and the words used are embarrassing and ashamed because we did something together before kick-off that every Premier League player and club has done, and it has been really powerful to see and watch, and I have been really proud of it.
"After the game, when we have got a beating from Man City that we are gutted about anyway, the real defeat was seeing that, and having what we felt was a unity and solidarity outshone by a minority."
Rodriguez, who started his career with his hometown club, before leaving in a then-record £7m switch to Southampton in the summer of 2012, returned in the summer from West Brom.
He was born in Burnley, and went to school at Heasandford Primary School and Barden High School in town.
He is, therefore, the best qualified in the dressing room to comment on whether the banner represented the feeling in Burnley: "I've always said I'm very proud of coming from Burnley, I respect everyone in the town, I've been brought up well, and I'm very proud that I've been brought up in a multi-cultural, multi-racial school.
"I've got good friends and I've learned a lot from different cultures and races, people from different places.
"I've been very fortunate to be able to learn and get to know people I wouldn't necessarily be able to if Ii didn't play football.
"I feel Burnley is a working class town and everyone sticks together, and in this sense we have stuck together and I feel we've been let down with what happened.
"We were gutted, and it's not what we're about, definitely not."
Burnley has been back on the map over the last few years, much due to the exploits of its football club in the most popular league in the world.
So it is a huge disappointment for the town's name to be tarnished in such a way: "It's sad, you look at how well it's gone with the Premier League, the clubs, the players, the staff - everyone together with Black Lives Matter, I think it's been really powerful and shows unity and strength, everyone comes together no matter what race, religion, culture.
"You are together as one. I feel that's really powerful.
"So to do it when we're showing our strength and unity to overcome racism, I think it was gutting and we were ashamed."
There has been resistance from people, insisting all lives matter, and white lives matter - although Black Lives Matter has not suggested only black lives matter, which seems to have been misunderstood by many.
Rodriguez added: "Every human being on the planet matters, we should be together no matter where you're from.
"With the terrible things that have happened, the years of persecution black people have gone through, this is what Black Lives Matter is for, to see it, to help and move forward together as one.
"For me, that's so powerful, more powerful than anything.
"And for it to happen on that stage, I feel it's just such a sad thing.
"I'm very proud of the club and players that we stuck to it and really got behind it, and we were absolutely gutted."
The dressing room want to maintain their support for Black Lives Matter, not just abandon the message after one round of Premier League games.
This is a moment of change, and the players hope to grasp it: "We need to continue this, not just for this moment, years to come, as long as it takes, until we get what we want, which is equality throughout.
"It's not going to spur us on more, because we're very much driven by the Black Lives Matter, but we will continue asking questions, learning and moving forward as one.
"The biggest and strongest message, to come together is so powerful, rather than having differences.
"That is the main thing."
So many people will tell sportsmen and women to 'stay in their lane', not to comment on politics, but Rodriguez hopes to help keep the message alive and help educate people: "Of course, you hope that you can affect it.
"We are role models and everyone is learning in life, you never stop learning.
"I hope that as role models that we can help people to learn and respect and move forward together. That is the most important thing.
"You look back and you learn through anything you can to understand.
"I feel that is part of it, to educate yourself and understand how difficult it is. I feel like for us it is really powerful that no matter what race you are or where you come from that you can come together and feel equal and that is the main thing, equality."
He added: "I feel that we are role models and human beings, regardless of what we do.
"People might see me as a footballer, but I am a human being and I feel like if you can help create a better life or anything that is going to be an improvement or to help people, then I don't see why you wouldn't help situations.
"I am very proud of everyone in the Premier League, the teams, staff and players that have come together to show solidarity.
"It is definitely important to keep doing that."
And Rodriguez is confident the stunt was played out by people who are in a minority in the town: "I don't feel like for me it (the town) does have a problem.
"We stick together in tough times and we come together and respect each other.
"I feel the majority of people do believe in what we are doing and it might be a small minority that think like that, but it is sad and we just have to carry on as one together."