The Saints had not lost at St Mary’s since the Blues won here in February. But Hazard gave the visitors the lead after only six minutes when he turned inside Steven Davies to drill home his fifth goal of the season – more than the Belgian scored in the whole of the last campaign.
Claude Puel’s side dominated possession but Thibaut Courtois’s only first-half save came from a Dusan Tadic free kick.
Costa was denied by Fraser Forster when sent through on goal by Nemanja Matic just before the break. The Spain striker made no mistake after 55 minutes when he was given time and space on the edge of the Saints box by Cuco Martina before curling his shot into the top corner.
Forster made futher saves from Hazard and Moses while Charlie Austin had a late effort disallowed for a borderline offside decision.