"Amy will be a fierce and formidable full-time addition to our GMA team," Sherwood said in a statement.
Elliott has accepted a position with NBC Sports. Sherwood said negotiations with Elliott had gone on for several months.
"In good faith, we worked hard to close a significant gap between our generous offer and his expectations. In the end, Josh felt he deserved a different deal and so he chose a new path," Sherwood said.
Elliott is the second person to leave "GMA" this month. Weather forecaster Sam Champion also departed the No. 1-ranked morning show in early March for The Weather Channel.
Robach has become quite familiar to the "GMA" audience, filling in for Robin Roberts as she recovered from a bone marrow transplant. Robach herself is battling breast cancer.
Robach has covered a variety of news stories, from the Sandy Hook school shooting, to the birth of Prince George and most recently the Sochi Olympics.
Robach got her start as a general assignment reporter in South Carolina. From there she became a morning anchor in Washington D.C., where she covered the 9/11 terrorist attacks from the Pentagon. She came to ABC from NBC, where she was an anchor and co-host for five years.
The Associated Press contributed to this report.