Southampton were undone early on as Hazard punished mistakes from Steven Davis and goalkeeper Forster. They started the season most impressively, briefly hanging on to Manchester City's coattails, but a run of five without victory has brought back a familiar sense of perspective.
Chelsea were denied a third by some last-ditch defending after Forster had spilled Moses' shot and Davis at last threatened Thibaut Courtois' goal when his header glanced off the top of the crossbar.
West Ham United manager Slaven Bilic is weighing up the option of starting fit-again Andre Ayew ahead of their English Premier League clash against Everton on Sunday.
However, Koeman said in his news conference that Barkley put in "one of his best [performances] of this season".
Everton veteran Gareth Barry, who has played over 600 Premier League games - fewer only than Frank Lampard and Ryan Giggs, admits he has to give Barkley some words of encouragement during a challenging time in his development.
Everton got the better of West Ham at Goodison Park to end their four-game Premier League winless streak.
Everton boss Ronald Koeman was pleased with his side's second-half performance in particular.
"It was a total team performance today", Koeman is quoted as saying by BBC Sport. "Ross Barkley had more impact in the last part of our attacking, not only in scoring but he was risky throughout".
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"He can be very good, but it's all about knowing what you need to do".
After a cautious start to the encounter by both teams, the match sprang to life in the 12th minute when Pedro Obiang fired over from the centre of the area after being teed up by Dimitri Payet.
And Bilic admits he remains a fan of Everton's tight ground - and not just because West Ham clinched a dramatic 3-2 win there last season.
Barkley still a lot to do but beat Adrian with an effort back across the West Ham keeper. The home side were excellent at Turf Moor last week with loads of attacking intent on display and you could see the anguish etched on Koeman's face after Burnley nicked all three points with a late victor.
Lukaku kept up his remarkable scoring run against the Hammers by heading the Toffees in front in the 50th minute, getting on the end of Yannick Bolasie's cross after Adrian could only parry a Seamus Coleman strike.
It remains to be seen how this episode will pan out with Ayew's commitment to his nation unquestionable while West Ham United had to smash their transfer record for his signature in the summer.
The result lifted sixth-placed Everton back to within five points of the top-flight summit, while West Ham remain 16th, three points above the relegation zone. "It was a good save from Adrian then we had a chance to clear, we didn't, we waited too long".