Bastianini had qualified third on the grid but took the lead from world championship contender Jorge Martin of Prima Pramac Racing on the sixth lap.
"The lap time was too crazy, the sprint has been crazy! It's my first victory of the year, I'm so happy. My pace was good. Here at Silverstone, (it) is one of my favourite tracks," Bastianini, who moved to third in the championship standings, said.
Martin had shot off the line from fourth to take the lead on the opening lap over pole-sitter Aleix Espargaro. With the podium, the Spaniard reduced the gap to Bagnaia to one point in the championship ahead of Sunday's race.
"It was really nice to race here in Silverstone. I was pushing like it was qualifying, but finally, Enea was super, was stronger than me... Hopefully, tomorrow we can bring it to the first position," Martin said.
Aprilia's Espargaro, who smashed the lap record in the final moments of qualifying at Silverstone to clinch his second pole position of the season, was under constant pressure from Martin before finishing in third.