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.
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.
That netted him a comfortable second place, albeit a distant 23 seconds behind the dominant Red Bull of Verstappen.
"What a result for our team, we definitely didn't expect to have the result we had today," added Hamilton.
"I just really want to take my hat off to my team, a big, big thank you to everyone back to the factory for continuing to push and bring us a little bit closer to the [Red] Bulls.
When asked by interviewer and former rival Nico Rosberg if Mercedes can still close the gap to Red Bull this year, Hamilton said: "I think they are too quick at the moment, but we're working at it.
"So, just one step at a time and if we can get close by the end of the year, that'd be awesome. But if not, next year."
After outscoring a struggling Aston Martin by 19 points in Barcelona, Mercedes has now jumped clear of its customer team to take second in the constructors' standings by 18 points.