The American boss replaced Francesco Guidolin in October but lasted just 85 days at Swansea, managed the club for 11 matches, and achieved a 2-2-7 record.
“We are sorry to lose Bob after such a short period of time,’’ confirmed Swans chairman Huw Jenkins. “Unfortunately things haven’t worked out as planned and we felt we had to make the change with half the Premier League season remaining.
“With the club going through such a tough time, we have to try and find the answers to get ourselves out of trouble.
“Personally, I have nothing but praise for Bob. He is a good man; a good person who gave everything to the job. His work-rate is phenomenal and we wish him well for the future.’’
First-team coaches Paul Williams and Alan Curtis will assume temporary charge of the club as they hunt for a new manager.