The world championship leader carried too much speed through the left-handed Turn 15 and hit the barrier.
Verstappen was unable to reverse away from the accident and the session was red-flagged with 30 minutes remaining.
Verstappen’s Red Bull team will now face a race against time to ensure his car is ready for qualifying, which takes place at 1pm.
Verstappen, who holds a four-point lead over Lewis Hamilton heading into Sunday’s race, suffered suspension damage in the accident.
After leaving his cockpit, Verstappen sat hunched over on a concrete block as he pondered just how big of a blow his practice crash might be ahead of the sixth round of the campaign in Baku.
In Verstappen’s absence, the AlphaTauri of Pierre Gasly finished as the surprise pace setter as he ended the running three tenths clear of Red Bull’s Sergio Perez, with Hamilton third.
However, the world champion benefitted from a mighty tow off the back of Perez’s Red Bull on the long drag to the finish line, with his Mercedes team still appearing to struggle for pace.
Indeed, Hamilton’s team-mate Valtteri Bottas finished only 13th, 1.5 seconds off the pace.
Without the slipstream, Hamilton would have been around a second slower than Gasly which would have put him ninth.
Charles Leclerc finished fourth for Ferrari ahead of team-mate Carlos Sainz with British driver Lando Norris sixth in his McLaren.
George Russell broke down in the closing stages following a mechanical failure.
Leading Times after Third Practice: 1 Pierre Gasly (Fra) Scuderia AlphaTauri 1min 42.251secs, 2 Sergio Perez (Mex) Red Bull 1:42.595, 3 Lewis Hamilton (Gbr) Mercedes GP 1:42.697, 4 Charles Leclerc (Mon) Ferrari 1:42.778, 5 Carlos Sainz Jr (Spa) Ferrari. 1:43.006, 6 Lando Norris (Gbr) McLaren 1:43.011, 7 Fernando Alonso (Spa) Alpine 1:43.080, 8 Yuki Tsunoda (Jpn) Scuderia AlphaTauri 1:43.244, 9 Esteban Ocon (Fra) Alpine 1:43.294, 10 Daniel Ricciardo (Aus) McLaren 1:43.557, 11 Sebastian Vettel (Ger) Aston Martin 1:43.585, 12 Lance Stroll (Can) Aston Martin 1:43.682, 13 Valtteri Bottas (Fin) Mercedes GP 1:43.745, 14 Kimi Raikkonen (Fin) Alfa Romeo Racing 1:43.826, 15 Max Verstappen (Ned) Red Bull 1:43.984, 16 Nicholas Latifi (Can) Williams 1:44.054, 17 Antonio Giovinazzi (Ita) Alfa Romeo Racing 1:44.113, 18 George Russell (Gbr) Williams 1:44.434, 19 Mick Schumacher (Ger) Haas F1 1:45.282, 20 Nikita Mazepin (Raf) Haas F1 1:45.711
Additional reporting by PA.