The 25-year-old Mathews was Jayawardene's deputy and was the heir apparent to the captaincy since the latter announced that he would be quitting captaincy after Sri Lanka's tour of Australia.
Wicketkeeper-batsman Dinesh Chandimal will be Mathews' deputy, but will lead the national side in the Twenty20 format.
Lasith Malinga will be the team's vice-captain for T20.
The selectors have also named a 20-member pool of players for the two-match home Test series against Bangladesh, starting next month.
Veteran Thilan Samaraweera, opener opener Tharanga Paranavithana and wicketkeeper Prasanna Jayawardene have been dropped from the squad.
Four uncapped players Kithruwan Vithanage, Ashen Silva, a medium pacer Dushmantha Chameera and off-spinner Tharindu Kaushal have been included in the Test squad for the series against Bangladesh next month.
Test squad: Angelo Mathews (c), Dinesh Chandimal, Tillakaratne Dilshan, Dimuth Karunaratne, Ashen Silva, Kumar Sangakkara, Lahiru Thirimanne, Mahela Jayawardene, Kushal Janith Perera, Jeevan Mendis, Kithuruwan Vithanage, Ajantha Mendis, Suraj Randiv, Nuwan Kulasekara, Shaminda Eranga, Chanaka Welegedera, Nuwan Pradeep, Dushmantha Chameera, Rangana Herath, Tharindu Kaushal.