The reigning Euroformula Open champion lined up second behind his Italian team-mate before taking the lead on lap 15 after Fornaroli began to struggle with his tyres.
Despite a second safety car and late rain, Goethe held on for his maiden series victory, with Fornaroli second and Pepe Marti in third for Campos.
Fornaroli led off the line, with Marti taking second from Goethe on the first lap as the pack tussled at Luffield.
Paul Aron (Prema) took fourth from Christian Mansell as ahead, Goethe reclaimed second at Stowe.
The first safety car came at the end of the first lap as several drivers tangled, with Luke Browning (Hitech) stopping on the Hamilton Straight after being nudged off by Campos driver Hugh Barter.
Caio Collet (Van Amersfoort Racing) also had contact with MP Motorsport driver Jonny Edgar, forcing both into the pits with damage.
There was confusion over who ran third between Marti and Aron, with both drivers claiming they should have the place, though Marti held position.
Racing resumed on lap seven before the yellow flags were thrown as Zak O’Sullivan stopped in his Prema-run car at Luffield, having been pushed wide by Kaylen Frederick (ART) before spinning, though he managed to get going again before pitting.
Dino Beganovic, who started last for Prema after his engine blew during qualifying, was up to 17th by lap nine.
Marti began losing time to the leading pair before Goethe took the lead on lap 15, with better pace than his team-mate, while Aron attempted a move on Marti at Stowe but fell back.
Rain began to fall on lap 17 as dark clouds loomed, but Goethe had managed to build a 1.5s gap ahead to Fornaroli, with a further 0.5s back to Marti.
The safety car was then deployed for a second time as Roberto Faria ended up in the barrier at Luffield in his PHM-run car after contact with Tommy Smith (VAR).
Racing resumed with two laps remaining, but Goethe maintained his lead to cross the line 1.7s ahead of Fornaroli.
With Marti completing the podium Aron finished fourth, with Saturday podium finisher Mansell (Campos) in fifth and championship leader Gabriel Bortoleto (Trident) in sixth.
Hitech driver Gabriele Mini crossed the line seventh, with sprint race winner Franco Colapinto in eighth. Collet and Gregoire Saucy (ART) rounded off the top 10.
F3 resumes in Budapest from 21-23 July.
F3 Silverstone - Feature race results
| Cla | Driver | Team | Laps | Gap |
|---|---|---|---|---|
| 1 | Oliver Goethe |  Trident |
22 | |
| 2 | Leonardo Fornaroli | Trident |
22 | 1.720 |
| 3 | Pepe Martí |  Campos Racing |
22 | 2.128 |
| 4 | Paul Aron | Prema Powerteam |
22 | 4.384 |
| 5 | Christian Mansell | Campos Racing |
22 | 5.968 |
| 6 | Gabriel Bortoleto | Trident |
22 | 7.334 |
| 7 | Gabriele Minì | Hitech Pulse-Eight | 22 | 7.997 |
| 8 |  Franco Colapinto |
 MP Motorsport |
22 | 8.529 |
| 9 |  Caio Collet |
Van Amersfoort Racing |
22 | 9.680 |
| 10 |  Gregoire Saucy |
 ART Grand Prix |
22 | 10.018 |
| 11 | Tommy Smith | Van Amersfoort Racing |
22 | 10.192 |
| 12 |  Sebastian Montoya |
Hitech Pulse-Eight | 22 | 11.758 |
| 13 | Nikola Tsolov | ART Grand Prix |
22 | 12.148 |
| 14 | Mari Boya | MP Motorsport |
22 | 12.700 |
| 15 | Hugh Barter | Campos Racing |
22 | 13.169 |
| 16 | Dino Beganovic | Prema Powerteam |
22 | 14.075 |
| 17 |  Rafael Villagomez |
Van Amersfoort Racing |
22 | 14.767 |
| 18 | Nikita Bedrin | Jenzer Motorsport |
22 | 15.496 |
| 19 |  Zak O'Sullivan |
Prema Powerteam |
22 | 15.808 |
| 20 |  Kaylen Frederick |
ART Grand Prix |
22 | 16.481 |
| 21 | Oliver Gray | Rodin Carlin | 22 | 18.067 |
| 22 | Taylor Barnard | Jenzer Motorsport |
22 | 18.914 |
| 23 | McKenzy Cresswell | PHM Racing by Charouz | 22 | 20.308 |
| 24 |  Sophia Flörsch |
PHM Racing by Charouz | 22 | 20.723 |
| 25 |  Ido Cohen |
Rodin Carlin | 22 | 22.913 |
| 26 | Alex Garcia | Jenzer Motorsport |
22 | 26.901 |
| Roberto Faria | PHM Racing by Charouz | 16 | ||
| Max Esterson | Rodin Carlin | 14 | ||
| Jonny Edgar |
MP Motorsport |
11 | ||
| Luke Browning | Hitech Pulse-Eight | 1 | ||
| View full results | ||||
