Norris and Hamilton stayed out during an early safety car, deployed after Aston Martin’s Lance Stoll crashed on lap six.
They stopped with 13 and 14 laps to go, Hamilton a lap ahead of Norris, and fitted soft tyres, hoping they could use their extra grip to get ahead.
But after initially taking a share of Beerman’s advantage, their tires began to dry up, and Norris and Hamilton were left battling each other for eighth place instead of challenging the youngster.
Beerman started 11th when drafted before the final practice and was 12th after the safety car.
He passed Zhou Gunayu’s Sauber and then Nico Hulkenberg’s Haas and then benefited from the tactics of Hamilton and Norris on a perfect start that confirmed him as a potential star of the future.
Biermann was named Driver of the Day in a fan vote and teammate Leclerc paid tribute to him after the race.
“He fully deserves it,” said Leclerc. “He did an incredible job. In P3 he was on the pace on the straight, in qualifying he did an incredible job and just narrowly missed out on Q3.
“To finish seventh in your first race in F1 in a new car is incredibly impressive. Everyone has realized how talented he is, and it’s only a matter of time before he’s in F1.”