The Blues took the lead through Hazard's 47th-minute penalty after Michail Antonio had felled Cesar Azpilicueta in the area before James Collins replied for the Hammers with a nicely hit left-footed volley following a goalmouth scramble.
In the 89th minute, Diego Costa came up with a quality finish from range to win it for the West Londoners, who according to Hazard played like 'tigers', allowing them to kick off the new campaign with all three points to start putting last season's debacle behind them.
"We played every minute like tigers," he said.
"I missed my last couple of penalties. Oscar wanted to take but I said no it is my turn. It is always good to score a penalty.
"Last season is finished. We are all looking to the future now. We want to carry on like this now."