Leading by Nathan Redmond's goal from the first leg two weeks ago, Claude Puel's side showed great resolve to survive a barrage of second-half pressure at Anfield before securing their 2-0 aggregate triumph when Shane Long struck late on.
Hazard held his nerve to slot home a last-minute penalty for a 2-2 draw vs West Bromwich Albion. Suarez scored two as Liverpool thrashed Fulham 4-0.
Frank Lampard quickly cancelled out Shane Long's early opener for the visitors at Wembley Stadium, and the lack of further goals meant the score between the sides finished 1-1.
West Brom were almost level in the 68th minute when Graham Dorrans crashed a free-kick against the crossbar. In a frantic finish, Steven Pienaar then hit the post for Everton before Moyes' men hung on for the three points.
West Ham threatened first, Matt Taylor heading over from Carlton Cole's left-wing cross, before the hosts took a grip on the game.
Chelsea coach Roberto Di Matteo paid a heavy price for tinkering with his team as the Blues went down 2-1 at his former club West Bromwich Albion on Saturday.
West Bromwich got back to winning ways with a battling 2-1 win over struggling Bolton at The Hawthorns on Saturday.
Republic of Ireland striker Shane Long has been ruled out of Friday's crucial Euro 2012 qualifier against Slovakia in Dublin after suffering a calf strain in training.