The Tech 3 Yamaha rider has enjoyed an excellent season to date, taking two podium finishes, and he continued that impressive form as a lap of one minute 34.398 seconds saw him beat off the Honda of Marc Marquez by three-tenths of a second.
The last British rider to take pole for a race in the top class was Jeremy McWilliams at the 2002 Australian Grand Prix.
Reigning world champion Jorge Lorenzo missed the session after breaking his collarbone in a heavy practice crash on Thursday, while Marquez was hindered by the broken finger and toe he suffered in an accident on Friday morning.
But the Honda rookie was still able to secure a front-row start, with Stefan Bradl doing a fine job to round out the top three on his LCR Honda.
Valentino Rossi was again fourth for the factory Yamaha squad, while championship leader Dani Pedrosa was fifth.
The Spaniard had led the way halfway through the session, but he opted not to go out again after a crash as he attempted to improve his time.
Crutchlow's team-mate Bradley Smith was an encouraging career-best sixth as he rides for the first time since undergoing surgery on injuries to his hand and wrist.
Aleix Espargaro's superb weekend continued as he was the top CRT machine in sixth aboard his Aspar Aprilia, ahead of Alvaro Bautista, team-mate Randy de Puniet and Ducati's Nicky Hayden.
The Ducati were enduring a miserable weekend, with Hayden's team-mate Andrea Dovizioso 15th.
Despite missing the session, Lorenzo is 12th on the grid thanks to his fast time in Thursday morning practice.
The Spaniard had been set to miss the rest of the weekend, but is due to fly back to Assen following surgery on his collarbone, meaning he could yet race if he is given medical clearance.