Amajimbos scored midway through the second half thanks to a penalty from Luke le Roux.
However, Tanzania's captain Ally Msengi responded from the spot soon after at Dobsonville Stadium.
The return leg of the tie will take place in Dar es Salaam on August 21.
Molefi Ntseki's charges entered the match on the back of a 3-1 loss on penalties to Namibia in the final of the COSAFA Under-17 Championship in Mauritius on July 31.
SA U-17 XI: Mavuso, Fleurs, Mokoena, Mazibuko, Shinga, Dladla, Mchilizeli, Radzilani, Baadjies, Monyane, Ntuli.