Russell was running in third place with Hamilton behind him when the pair suffered a water leak, allowing Ferrari's Carlos Sainz to pass the seven-time world champion in the penultimate lap.
The issue also pushed Russell onto the defensive, but he held off the Spaniard to take the third podium spot, his second of the season after Australia, albeit 20 seconds off Sergio Perez in second.
Like Russell, Hamilton was told to lift and coast with the potentially race-ending issue, but managed to finish in fifth.
Russell said: "Yeah, it was very difficult in the last few laps. It was a survival race when we knew we had a gap.
"As I said, just proud to bring it home in P3 and thanks to all the fans here. They've been amazing all weekend and the support for F1 at the moment, it is in an incredible place, so it's great to be here at the moment."
He said overheating issues "hurt us a lot, to be honest," adding: "When I had the Red Bull was in my mirrors, all I was doing was my maximum to keep him behind and it was an enjoyable, good couple of laps.
"Just pleased to be able to bring home P3, a lot of points on the board for us."
Despite the strong result, the team still sits third in the constructors' championship, 75 points adrift of leaders Red Bull and 49 behind second-placed Ferrari.
Both drivers felt the team had made a big step forward to solve its porpoising problems thanks to the updates brought to Barcelona.
But Russell was reluctant to say the team had returned to its usual form, adding that the race was "very tough."
He said: "I'd love to say that but today was very tough. I gave everything I could to hold Max off, he did an amazing job. And so proud to be standing here.
"We've had to work so hard. So this is for everyone back at Brackley and Brixworth. So thank you."
