The Mercedes driver, starting from his maiden F1 pole position, opted to begin the Hungary race on soft tyres while the Ferraris directly behind him picked the mediums.
With a heavy rain shower forecast for the start of the race, Russell said he was "rubbing his hands together" in anticipation to use the soft tyre advantage which are better suited to run in slippery conditions.
But once the rain didn't materialise, he was kept in check by the Ferraris and forced to pit earlier, leaving him exposed on the mediums later in the race.
"It was always going to be a tricky afternoon because there was rain in the air throughout," Russell said, having finished third. "Usually when you are on pole position and it is spitting you are pretty disappointed but knowing that I was on the soft tyres I was kind of rubbing my hands together.
"I made a really good start, the first four laps were really strong and pulled the gap to three seconds to Carlos [Sainz] and unfortunately the rain stopped so the medium tyres came into their own but we still had a good first stint and we pitted early on both occasions to cover Max and cover the Ferraris and ultimately Max was too fast for us today.
"That put us in a very tricky position in that last stint on the mediums and I was already struggling a bit and my engineer said 25 laps to go and the rain started falling and temperatures were dropping so I thought it was going to be a tricky last few laps."
Having been leapfrogged by Verstappen in the second round of pitstops, Russell then conceded position to Mercedes teammate Lewis Hamilton who was on the soft tyres for the final stint to drop to third by the finish after keeping clear of Sainz.
"Lewis did an amazing job and had far superior pace at the end, he had a very strong middle stint," Russell said.
"I think as a team we've got a lot to be proud about, we did everything right for both of us, to give us as a team the best opportunity to fight for victory. Unfortunately it didn't go our way today but as a team we did everything right."
Russell has moved up to fourth place in the F1 drivers' standings, two points ahead of Sainz, thanks to his fifth podium of the season.
