Hazard scored to take Mourinho's unbeaten home league record at Stamford Bridge to 66 matches following 60 unbeaten games in his first spell there between 2004 and 2007.
Samuel Eto'o put Chelsea ahead but goals from Shane Long and Stephane Sessegnon put West Brom, managed by former Chelsea assistant manager Steve Clarke ahead and within sight of victory until the late drama.
Although Chelsea blew the chance to close the gap on leaders Arsenal to two points, Liverpool made no mistake crushing a woeful Fulham 4-0 at Anfield.
Luis Suarez scored twice to take his goal tally to eight in the seven matches he has played since returning from suspension.
Arsenal, who play Manchester United at Old Trafford on Sunday, top the table with 25 points from 10 matches, followed by Liverpool on 23, Southampton on 22, and Chelsea on 21.
Southampton moved into third place with a highly impressive 4-1 win over Hull City with Morgan Schneiderlin, Rickie Lambert from the spot and Adam Lallana with a superb individual goal, putting Saints 3-0 ahead after 37 minutes. Steven Davis added the fourth in the 88th minute.
In the other afternoon matches, Aston Villa ended a run of seven and a half hours without a Premier League goal when Leandro Bacuna scored after 76 minutes to give them the lead against Cardiff City.
Libor Kozak then added a second in the 83rd minute to give Villa a 2-0 victory.
Bottom-placed Crystal Palace ended a run of six successive defeats when they drew 0-0 at Selhurst Park with Everton.
Norwich City were meeting West Ham United in the late kickoff (1730GMT).
(Reporting by Mike Collett, editing by Alan Baldwin)
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