The midfielder, 33, had fallen out of favour with the Rams and had been linked with a player-coach role at the Blue Square Premier outfitWrexham before opting to head to the South Coast.
Savage joined Derby in January, having previously enjoyed successful stints with Blackburn Rovers, Birmingham City, Leicester City and Crewe Alexandra.
"We're delighted to have Robbie here. It means we have even more competition for places in the side," said the Seagulls manager Micky Adams.
"He is a terrific character on and off the pitch and, having worked with him before, I know the sort of quality he'll add to the team."