Alonso then hit his team-mate Lance Stroll while Norris was around the outside, but could do nothing to avoid the chaos and was hit by the Aston Martin, sending him spinning into the outfield.
The McLaren was unable to get going again and was forced to retire from the race along with Stroll.
The race was halted by a safety car until the track marshals could safely remove Norris' stricken McLaren.
Despite the Brit retiring through no fault of his own in the crash, he was under investigation for entering the track on foot after exiting his car.
Stewards later gave him a reprimand, and fining him €50,000 with €25,000 suspended for the rest of the 2024 season.
The stewards commented: "During the hearing the driver of Car 4 realised that the situation could have been very dangerous for him.
"The Stewards reinforced the fact that crossing a live track can cause extremely
dangerous situations and the drivers have to be very cautious about it."
Piastri ended up finishing sixth in the Sprint, but a penalty for Norris would mark a disappointing weekend thus far in Miami, despite bringing a significant amount of upgrades to the car.
The McLarens had looked good for pole position for the Sprint on Friday, but Oscar Piastri and Norris could only manage sixth and ninth respectively.