The Arsenal forward returned to Juan Antonio Pizzi's starting lineup after being on the bench for the 2-0 win over Cameroon in their first fixture of the tournament.
Mustafi's Gunners teammate made the most of the blunder by playing a smart one-two with Vidal before bearing forward and firing a left-footed shot beyond goalkeeper Marc-Andre ter Stegen.
Sanchez's goal was his 38th in Roja colours, eclipsing the record of 37 set by Marcelo Salas in 2007.
However, the Chileans were then furious as Mustafi clattered through the back of Vidal later on in the first half, implying that he should have seen red.
Germany had their backs to the wall for the majority of the opening half but they nonetheless drew level in the 41st minute.
The result means that the two nations need just a point each in their next games to guarantee qualification from Group B.