Antonio Conte's side produced a vibrant display to inflict a first home defeat of the season on Southampton and underline their title credentials.
Hazard cut back from a tight angle and rifled a shot through Fraser Forster's legs after six minutes to give Chelsea a lead they never looked like relinquishing against a Southampton side who could do nothing with their superior possession.
Hazard forced Forster into another save before halftime while he also kept out Costa as Chelsea continued to look dangerous on the break.
Spain striker Costa doubled the lead 10 minutes after the break with a sensational strike, receiving a pass from Hazard and beating Forster with a swerving right-foot shot from 25 metres to take his league tally for the season to eight.
Chelsea, who moved back above Tottenham Hotspur into fourth spot just one point off the leaders, had the opportunities to increase their tally with Costa sqaundering another glorious chance when he elected not to shoot.
Southampton's best chance fell to Charlie Austin who headed over the bar.
Chelsea have now amassed double the amount of points (22) that they managed after 10 games of last season when their hopes of defending their title under Jose Mourinho were already in tatters.
"Southampton's record at home is very good -- we knew that. But we knew if we carried on our form, we'd cause them problems. We worked hard," Chelsea defender Gary Cahill told Sky Sports.
"It's too early in terms of any team talking about winning the league but its important to stay near the top."
(Reporting by Martyn Herman; Editing by Ian Chadband)
