Toro Rosso's Sainz left the track at Turn 13 during third practice and his car ended up lodged underneath the safety barriers.
It took several minutes for the 21-year-old Sainz to be extricated from the wreckage to receive medical treatment and the session was brought to a hasty conclusion.
Sainz, the son of former world rally champion Carlos Sainz, was taken to a hospital about 30 km from the circuit but escaped serious injury.
Pictures released on social media showed him giving the thumbs up and he was later released from hospital.
He has been given medical clearance to start in Sunday's race, but it was unclear if he would take his place at the rear of the grid.
Coming just over a year after an accident at the 2014 Japanese Grand Prix which eventually claimed the life of French driver Jules Bianchi, the relief in the F1 paddock at his fortunate escape was palpable.
But questions were being asked why his car, now fitted with a lower nose cone, was able to go under the tec-pro barriers, designed to absorb impact of a crash at top speed.
Ferrari ace Sebastian Vettel voiced the opinion of many: "It was shocking to see he was so deep in the barriers and he was covered by them too," he said.
"The idea is not for the barriers to come on top of you or the car go under, so it is something we need to understand.
"I don't know why if there was anything wrong with the way they installed the tec-pro barriers or if was just coincidence the way he crashed into them. I am sure that is something that needs to be avoided."
Sainz had been traveling at about 200mph (320kph) when he lost control of his car, glancing off the outside wall at the Sochi Autodrom before plowing straight into the barriers.
His father, competing at a rally in Morocco, was able to report good news after the initial fears.
"Just talked to Carlos. Thanks to God, he is ok. More testing going on, but much happier now. Thanks to everyone for the support," he said.
Sainz's team, a feeder for four-time constructors' champion Red Bull, also issued a statement soon after the accident.
"Following Carlos Sainz's accident during FP3 today in Sochi, the driver is perfectly conscious and was able to talk to the extrication team," it read.
Sainz was not the only driver to leave the track as drivers struggled to get to grips with the circuit, new to the Formula One roster in 2014.
Defending champion Lewis Hamilton also went off at Turn 13 in his Mercedes during the final qualifying session, leaving the way clear for his teammate and nearest title rival Nico Rosberg to take pole.
"I'm very happy, it's been a difficult weekend, but I found a good balance on the car," said the German.
Hamilton, 48 points clear of Rosberg in the championship standings, was second quickest with Valtteri Bottas a fine third for Williams.
Ferrari pair Vettel and Kimi Raikkonen were fourth and fifth fastest with Sainz's teenage teammate Max Verstappen in ninth.