Mr Carpenter had put his leadership on the line saying the ALP had to choose between him and Mr Burke.
The premier says Mr Burke has now agreed to go.
The ultimatum came after Mr Carpenter dumped Small Business Minister Norm Marlborough from cabinet after the Corruption and Crime Commission revealed phone records and tapped phone recordings between him and Mr Burke.
During the conversations the minister agreed to appoint a woman to a board at Mr Burke's urging and he took instructions from Mr Burke about how to respond to questions in parliament.
Mr Burke was jailed in 1994 for travel expense rorts but controversy has surrounded his involvement in Labor politics since Labor returned to power in WA in 2001.