After qualifying a disappointing 12th, Russell confirmed Mercedes' improved race pace from Friday practice to move up to third behind winner Max Verstappen and team-mate Lewis Hamilton.
Russell and Hamilton used their superior tyre management, which allowed them to avoid the hard tyre on their two-stopper, to defeat Ferrari's Carlos Sainz, who dropped from second to fifth.
According to Russell Mercedes' double podium shows that the Brackley team is finally on the right track after introducing comprehensive upgrades in Monaco, which were further exploited at the more traditional Barcelona layout.
"I'm a little bit surprised, for sure," said Russell, who finished eight seconds behind Hamilton.
"Kudos to the team for giving me a really great car today. A strong race for us as well from P12 coming all the way to P3.
"A sign of things to come hopefully for us as a team."
Hamilton, who briefly fell behind Aston Martin's Lance Stroll at the start, passed Sainz on track after the first round of pitstops and was never threatened again for second, using his softer mediums to distance the Ferrari.
Source: Autosport