The result means Amajimbos exit the competition with a 3-1 aggregate defeat, following their 1-1 draw at home two weeks ago.
Moelfi Ntseki's troops endured a tough time out in Central Africa, as the hosts bagged two goals without response on either side of half-time.
Tanzania found the opener through Mohamed Abdallah, who sent a shot beyond goalkeeper Glen Baadjes, in the 35th minute.
South Africa were handed a lifeline in the 45th minute as Ally Ng'anzi was sent for an early shower following a second yellow card.
However, despite the numerical advantage the visitors could not make their efforts count and were punished from a 84th minute counter-attack, as the home side extended their lead and put the tie beyond South Africa's reach.
Amajimbos join Nigeria on the outskirts as the third round qualifiers will decide who makes it to the 2017 U-17 Africa Cup of Nations in Madagascar.
The top four sides at the continental championships win qualification to the 2017 U-17 FIFA World Cup, which will be played in India between September and October next year.