Sri Lanka Cricket said the 38-year-old left-arm spinner Herath was named captain although he had retired from the shorter formats of the game.
"He has taken on the challenge of leading the team through the two Test series against the Bangladeshi team," the cricket board said in a statement.
It did not said why Mathews was dropped from the Test squad, but he returned home last month half way through the South African tour after twisting his ankle.
Herath had filled in during Sri Lanka's tour of Zimbabwe in November when he led his side to win both Tests.
Sri Lanka Cricket is expected to announce the full squad for the Bangladesh Tests on Tuesday. The first Test begins in Galle on March 7 while the second is in Colombo and starts on March 15.
Sri Lanka has a good record at home and beat Australia 3-0 in the last series that they hosted in August 2016.
Bangladesh by contrast have only won three of their Test matches on the road since gaining full status 17 years ago although they have shown recent signs of improvement and beat England in a home Test late last year.
The last series between the two sides in Sri Lanka in 2014 saw the hosts win one of the Tests by 248 runs while the second was a draw.