Holland followed in the footsteps of Leeds brothers Alistair and Jonny Brownlee to win Yorkshire’s 13th medal of the Olympic Games.
Both Holland and Stanford were in medal contention throughout, cycling among the pack after the swim before running side by side as leading pair Gwen Jorgensen and Nicola Spirig broke away.
It was 30-year-old Holland who had more left in the tank in the final 100m and completed the podium, with American Jorgensen winning gold ahead of Spirig.
Holland said: “I have such mixed emotions. I was absolutely delighted when I crossed the line and I still am. To come and win a medal for Team GB and for myself is absolutely what I came here for.
“But to have to beat out your best friend, your training partner, your housemate is hard. And I guess about 5km into the run, I knew that it was going to come down to me or Non for a medal.
“We’ve always said it’s fair game when it comes to the run. We knew we wanted a medal and we didn’t want to let Barbara Riveros [of Chile] back in from behind. So at that point we just had to keep the pressure on and keep running for the bronze.
“I wanted both of us to do it. Non is a huge part of what I do. Half of this medal is hers. I wouldn’t be the athlete I am if it wasn’t for her. I moved in with Non at the end of 2013 and I’ve become an exponentially better athlete for it.
“She is a huge part of what I do and I have to give so much credit to her. I know that she’s so strong and she’ll be back in Tokyo in four years’ time and I wouldn’t bet against her. The first thing I did when I regrouped myself was try to find her and I told her I’m so sorry.
“But of course she was completely gracious as she always is. She’s a total star and I know she is delighted for me and I understand that she’ll be gutted. Fourth is a horrible place to finish, especially at the Olympic Games.”