Rock-bottom Bournemouth, without seven long-term injured, can have no complaints about the result.
Brighton had to soak up plenty of pressure in the first half but Bournemouth's only real chance fell to Jo Kuffour who was unmarked when he headed over from 12 yards.
Brighton changed the game 17 seconds after the restart when Cox lashed home a right-foot shot after Savage had found a gap down the right.
Bournemouth's confidence drained with the setback and it was no surprise when Brighton scored again in the 66th minute.
Savage appeared to be the last player to touch the ball in from close range after Nicky Forster had broken down the right and set him up with an easy chance.
Bournemouth failed to muster a clear opening and Brighton goalkeeper Michels Kuipers had only one save to make, a straightforward header from Lee Bradbury.
At the final whistle the home crowd booed the players and manager Kevin Bond for another failure at their Dean Court ground.
The win was Brighton's fourth of the season but Bournemouth had a moment of bad luck when a Seagulls defender appeared to handle the ball in the area as Danny Hollands tried to force a way through.
Cherries striker Bradbury was booked in the 69th minute for pushing a defender and Savage, scorer of one goal and maker of the other, was also cautioned for retaliation.