Johnson's initial tackle on Zorko slipped high and as the umpire blew for a free kick, the Dockers youngster dumped his opponent into the ground.
Zorko, whose head hit the turf in the tackle, was awarded a free kick and had a shot on goal from the resulting 50-metre penalty.
The match review officer charged Johnson with rough conduct, grading the incident as careless conduct, high impact and high contact.
Johnson's ban is a further blow for the struggling Dockers, who have slumped to a 2-5 start to the season following the 48-point defeat to Brisbane.
West Coast's Sam Petrevski-Seton was also hit with a rough conduct ban, suspended for one match over his tackle on Carlton's Alex Cincotta.
The incident was assessed as careless, medium impact and high contact.
Petrevski-Seton's ban causes the injury-hit Eagles further pain after they fell to last spot on the ladder with their 108-point thrashing from the Blues.
GWS midfielder Xavier O'Halloran was fined $1500, which can be reduced to $1000 with an early guilty plea, for engaging in a melee and wrestling in the thrilling derby win over Sydney.
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