Lance and his team made a breakthrough in finding the optimal balance on the AMR21 to help make progress, while team-mate Sebastian Vettel endured a challenging race, including a spin and retirement.
The points proved crucial in an increasingly close Constructors’ Championship battle ahead of the Hungarian Grand Prix that precedes the summer break.
Lance started the Grand Prix from 14th, courtesy of a battling Sprint performance on Saturday, lining up on a set of used Medium tyres.
All drivers had a free choice of starting tyre, and 19 of the 20 drivers opted for the mid-range compound, meaning overtaking and strategic gains were more challenging.
But Lance seized the initiative with a strong launch off the line, passing George Russell, Kimi Raikkonen and Pierre Gasly before the end of the opening lap.
Get the full story from the British Grand Prix and go inside Lance's climb up the order. Dive into the facts and figures behind the Silverstone race and hear from Sebastian, Lance and Otmar.
His moves were well-timed because the race was halted after a collision between Lewis Hamilton and Max Verstappen up front.
Under the subsequent red flag, the team switched Lance to a newer set of used Medium tyres, allowing him to extend his first stint when the race resumed.
After a delay of almost 20 minutes, Lance made the most of a second standing start to gain another two places.
Swiftly dispatching Esteban Ocon, Lance then gained a place when Sebastian was caught up in an incident.
"The key to the race was making two strong starts either side of the red flag, where we gained several places each time," said Lance.
"The strategies were quite similar for the whole grid, making overtaking more tricky, but those starts set us up for a good race."
It set up a mid-race duel with Fernando Alonso, and Aston Martin Cognizant Formula One™ Team took the initiative by undercutting Alpine at the first pitstop on lap 23 of 52.
When the Spaniard finally stopped, Lance went wheel-to-wheel with Alonso on his newly-fitted used Hard tyres.
With strategies closely matched, and Lance in the middle of a DRS train, the Canadian focused on protecting his position late on.
He put up a strong defence against Red Bull’s Sergio Perez at Club corner and later held off Gasly for eighth.
Alongside AlphaTauri’s Yuki Tsunoda, Lance made up the most positions on track by gaining six places.
We have been working to find the right car balance and we had a breakthrough today, which is a great credit to the team and all the effort everyone has put in.
"I am really happy with my race because we were able to turn around a tricky start to the weekend by finishing in P8 and scoring four points," added Lance.
"We have been working to find the right car balance and we had a breakthrough today, which is a great credit to the team and all the effort everyone has put in."
Sebastian darted past Alonso at the beginning of the Grand Prix to move into seventh, keeping his cool when he ran three-wide with the Alpine driver and Carlos Sainz Jr at Aintree.
A strong exit out of the corner allowed him to hold onto seventh and pull a gap down the Wellington Straight.
The collision between Verstappen and Hamilton lifted Sebastian into sixth before the Red Flag was called.
Sebastian held sixth on the restart, going wheel-to-wheel with Alonso across the first half of the lap until a spin at the exit of Luffield.
The weekend was looking promising and, without all the problems, I should have finished well inside the points.
"I am very disappointed to lose the car in a wheel-to-wheel battle with Fernando [Alonso]," said Sebastian.
"I am not sure exactly what happened, but I misjudged it, lost grip, and spun. After that, I was right at the back."
An engine overheating issue prevented Sebastian from making up significant ground and curtailed his race on lap 40.
"The weekend was looking promising and, without all the problems, I should have finished well inside the points," added Sebastian.
We’re just getting started. Sign up to be the first to hear about latest news, new apparel drops and gain access to exclusive experiences.