"The work of marshals is not always seen, but it is vital to our sport and without their commitment, time and dedication, there would be no motorsport," Red Bull's race winner in Canada Sebastian Vettel said on his personal website.
"I am very, very sad to hear this news and my thoughts are with his family and friends," added the German.
The marshal, as yet unnamed, died in hospital from injuries sustained while removing the Sauber of Mexican rookie Esteban Gutierrez, who crashed out seven laps from the end of the race.
The FIA governing body said in a statement that the track worker had dropped his radio and attempted to pick it up. In doing so, he stumbled and was hit and run over by the recovery vehicle.
The marshal was the third to die at a grand prix since the turn of the century but the first since Graham Beveridge was killed by a loose tyre that flew through a gap in the safety fence at the 2001 Australian Grand Prix.
Italian fire marshal Paolo Ghislimberti died when he was hit by a wheel at the 2000 Italian Grand Prix at Monza, prompting a review of safety at race tracks and the introduction of improved measures to protect circuit workers.
Every Formula One ticket carries a warning that motorsport is dangerous but the fact that it had been more than a decade since the last fatality, despite marshals performing some of the riskiest roles, is testimony to the constant striving for safety.
The last driver fatality at a race weekend was Brazilian triple champion Ayrton Senna at Imola in 1994.
"My dearest condolences to the family of the marshal who lost his life today, our prayers for him & his family. RIP," Gutierrez wrote on his Twitter account.
"Shocked & saddened by the news that a marshal who is there to keep us safe has lost his life during our race today. Rest in peace my friend," said McLaren's Jenson Button, winner in Canada in 2011.
"Today there is nothing to celebrate. Terrible news arrive with the death of a marshal this race. Very sad. R.I.P," added Ferrari's Fernando Alonso, the double world champion who was runner-up to Vettel on Sunday.
While the marshal's death appeared a freak accident, there has long been concern about the use of such cranes while the race is still going on - but more from the point of view of driver safety.
Former racer Martin Brundle, now a commentator with Sky, observed during Sunday's grand prix - before the accident with the marshal - that it might have been safer to leave the car where it was rather than hurrying to remove it.
Now-retired seven-times world champion Michael Schumacher was fortunate to escape serious injury in Brazil in 2003 when a recovery crane was deployed during a rain-hit race and his Ferrari skidded off and almost crashed into it.
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- Death & Funeral
- Esteban Gutierrez