Lance had run well inside the top 10 for the majority of the Sochi race, while Sebastian initially rose up the order until the rain intensified and forced a late switch to Intermediates.
Lance launched strongly off the line at the start of the Grand Prix, jumping the Alpine of Fernando Alonso.
He then moved clear of Daniel Ricciardo's McLaren at Turn One, before taking advantage of Lewis Hamilton getting boxed in a field of cars to go fourth.
Alonso would run wide at Turn Two and re-join through the escape road ahead of the AMR21, but Lance stormed around the outside of the long Turn Three bend to remain ahead in fourth.
Lance then spent the opening 12 laps well within DRS range of George Russell, but the lessened impact of DRS at Sochi made it difficult to make a move for third place.
The team made the decisive call to bring Lance in and switch from his new Medium tyres to a set of new Hards, undercutting Russell in the process.
Russian Grand Prix
Uncover the story of the Russian Grand Prix with our in-depth race report. Discover the data underpinning our race and hear from Lance, Sebastian and CEO & Team Principal Otmar Szafnauer as they review the Sochi race.
Lance made the most of the clean air to punch in the fastest lap of the race on lap 15 of 53 and put himself firmly in contention for a strong result.
But the DRS train ahead made managing the Hard tyres all the more difficult in a likely long second stint, before the arrival of rain turned the race on its head.
Rain first fell at Turn 10 with less than 10 laps remaining and had drivers evaluating slicks or Intermediate tyres and Lance made the decision to stay on slick tyres.
Unfortunately, the rain intensified and Lance fell from a comfortable top 10 finish into the fringes of the points.
He switched to Intermediates for the final five laps and pushed hard to regain places, but ultimately finished 11th.
“In hindsight, stopping one lap earlier for Intermediates could have changed our race, but it is always hard to judge when the conditions are evolving.
"It is a real shame that the change in weather cost us a strong chance of a good result and plenty of points," said Lance.
"In hindsight, stopping one lap earlier for Intermediates could have changed our race, but it is always hard to judge when the conditions are evolving.
"I made the call for us to stay out and try to make it to the end, so it is down to me.
"It is even more disappointing considering we made a great start and got up to fourth in what was a really enjoyable battle in the first few corners."
Sebastian's start to the Grand Prix proved more challenging when Kimi Raikkonen passed him at the start and joined a DRS train led by Alonso all the way up in eighth.
While the four-time World Champion was always in contention to regain 10th, the lack of powerful DRS effect and limited overtaking opportunities hampered his first stint.
"In the early part of the race it was not easy to make much progress in the train of cars," added Sebastian.
The team and Sebastian extended the first stint to 26 laps, allowing him to gain places as rivals' completed their pitstops.
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Several slower stops elsewhere also contributed to his rise, running in eighth before pitting to switch from new Medium tyres to a set of brand-new Hards.
Unfortunately, Sebastian would again be hindered by a DRS train and circulated on the outskirts of the top 10 – until the rain arrived.
Sebastian opted to stay out on slicks and showed expert car control to rise all the way to sixth as rivals slid off the track or opted to pit for Intermediates.
But as the rain worsened, Sebastian fell down the order and was forced to pit two laps from the end for the Intermediate tyre.
He crossed the line 12th behind Lance as the weather twist hampered the team's pursuit of points.
When you have rain late in the race it can give you a big opportunity, but it did not go our way today.
"It felt like a 50/50 call as to whether we should pit for Intermediate tyres," said Sebastian. "It felt like the rain would stay light, so we stayed out, but then it became much heavier and caught us out.
"So we had to pit in the end and fit Intermediates, which dropped us down the order.
"When you have rain late in the race it can give you a big opportunity, but it did not go our way today."
The race also marked Lewis Hamilton's 100th Grand Prix victory, and CEO & Team Principal Otmar Szafnauer praised the Mercedes driver on his remarkable achievement.
"On behalf of all of us at Aston Martin Cognizant Formula One™ Team, I would like to say a big congrats to Lewis [Hamilton]," he said. "To win 100 Grands Prix is a truly extraordinary achievement. Well done!"
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