Garcia was elected to head the investigative branch of FIFA's newly-reformed ethics committee while German judge Hans-Joachim Eckert was appointed to head the tribunal which will judge cases and hand out sanctions.
"He [Garcia] will have not only the right but the duty to have this case analysed on ethics and moral grounds and then to report back to the executive committee," FIFA president Sepp Blatter told reporters.
"The chairmen of both chambers are totally independent, this had been requested by FIFA's Congress," added Blatter.
The bribes were paid by FIFA's former marketing partner ISL which collapsed in 2001.
Havelange is FIFA's honorary president while Teixeira quit as head of the Brazilian Football Confederation (CBF) earlier this year.
