The pair have won a medal at every competition they have entered since making their debut together earlier this year.
They won a World Cup bronze in February in Brazil and four World Series medals before a European silver in London, but winning an Olympic medal is the highlight of their new partnership.
However, they had to endure a nervous wait to see if they had made the podium in Rio and secure Great Britain only their third diving medal in 56 years.
Daley, 22, and 19-year-old Plymouth Diving’s Goodfellow needed to score more than 83 points for their final dive to claim a medal.
They had to wait what seemed like an eternity for their score to be announced but they jumped for joy when it was confirmed they had scored 88.64 points to beat highly experienced German pair Patrick Hausding and Sascha Klein to bronze.
As expected, China’s Aisen Chen and Yue Lin won with 496.98 points at the outdoor venue.
America’s David Boudia and Steele Johnson were second with 457.11 points.
Daley, who won an individual bronze in London 2012, said afterwards: “It was an agonising wait that felt like it was an eternity. We were stood there and knew we had done a good dive but were didn’t know if we’d done enough.
“It was just one of those moments where I knew how I would feel if it came up on the scoreboard that we finished fourth and I knew how I would feel if we finished third.
“We were just waiting and waiting with replay after replay, then all of a sudden the scores came up, I pounced on Dan and I don’t think he was quite ready for it and then before we knew it we were back in for a top bombing seventh dive.
“We knew we had a shot at winning a medal. We’ve only been diving together since October so we’re a new partnership and we didn’t actually expect to qualify for the Olympics.
“But since then we’ve won medals in every world event, World Series, European Championships. Back in the European Championships it was a similar situation going into the last round dive and the Germans pipped us by two points.
“We didn’t want to let that happen again. I said to Dan afterwards, don’t worry we’ll get them at the Olympics. And we did.”
Goodfellow, who paid tribute to his Plymouth-based coach Marc Holdsworth, said: “I wasn’t really thinking about Europeans in that last dive. I was just trying to block everything else out.
“I knew before we started that we’d be number eight and I knew no matter what situation we were in there would be loads of pressure going into it.
“So I just accepted that and tried to block everything else out and focus on my dives.
“Our fourth dive wasn’t amazing but towards the back end of our list we have a really high degree of difficulty.
“I knew our last two are our two real big ones and we’d have to nail them.”
Daley will dive in the individual event later in the Olympics. That event starts on August 19.
He added: “I’ve taken the pressure off myself now. Now I can go into that individual competition in the best frame of mind possible.
“It’s given me a taste of it and I want to go and get some more.
“Now I want to go and get a gold medal.”
Tomorrow evening, Plymouth’s Tonia Couch and Hudderfield teenager Lois Toulson will be hoping for success in the women’s 10m synchro event.