The ban means Lee will miss the first three weeks of the Scarlets' PRO12 campaign as well as friendlies against Bath and Gloucester.
Lee, who was making his first start for Wales, was cited for the incident which resulted in Van der Merwe receiving treatment for a blood injury.
IRB disciplinary officer Christopher Quinlan imposed an initial 10-week suspension. But the ban was reduced because of Lee's admission of reckless conduct, his youth and his remorse at the hearing.
Citing commissioner James Absalom from Kenya cited Lee under law 10.4 (a) - striking or punching an opponent.
Lee was brought into the starting XV by Warren Gatland ahead of the second Test as a replacement for the injured Adam Jones.
Wales lost the two-match series in agonising fashion after South Africa claimed a 31-30 victory on Saturday after prevailing 38-16 in Durban a week earlier.