Both traded punches in a maul and were then sent off in the second half of the Bulls' 19-16 victory away in Durban on Saturday.
They both pleaded guilty to the offence, but while Van der Merwe had the customary six-week sanction cut in half, the 37-year-old Brits must serve an extra week because he has two previous suspensions for fighting, SANZAAR said on Tuesday.
Brits and Van der Merwe are likely rivals for a spot in the South Africa squad for the World Cup in Japan later in the year, with Malcolm Marx and Bongi Mbonambi ahead of them in the pecking order.
(Reporting by Mark Gleeson; Editing by Sudipto Ganguly)
