Police in Calgary say the chase began when the minivan took off after colliding with a sedan at an intersection early Saturday.
The driver of the sedan gave chase and eventually confronted the minivan at a traffic light.
Police say the sedan driver, who is 28, reached into the van and hung on when it sped away.
Witnesses have told police the minivan attempted to swerve around a truck, then veered across several lanes of traffic onto a sidewalk with the sedan driver hanging out the window.
Police say the sedan driver suffered critical, life-threatening injuries when he was caught between the van and the wall of a building.
The van came to rest on the sidewalk and its driver, who is 43, was taken into custody.
Police are warning people against taking unnecessary risks is such situations.
"While the Calgary Police Service recognizes the heat of the moment in this circumstance, we would like to take this opportunity to remind people to always be safe," police say in a news release.
"It may be more prudent to call police with whatever information you have and let us deal with the offender rather than exposing yourself to the risk of serious injury or worse."
There's no word, yet, on any charges.
Police are asking anyone who may have seen either of the two collisions to contact them.