Three-goal thrashings at the hands of Bournemouth and Watford saw odds slashed on Conte becoming the latest Chelsea boss to be dismissed just nine months on from winning the Premier League.
However, Conte's name was sung loudly around Stamford Bridge as the brilliance of Belgian Hazard either side of a second-half strike from Victor Moses moved Chelsea back into the top four.
"I think they are reading speculation, rumours on me, and for sure I am grateful to the fans for supporting me this way," said Conte.
"Maybe in this year-and-a-half they understand my patience and will to defend this colour, this shirt, this badge."
Chelsea climbed into fourth with the gap between Manchester United in second and fifth-placed Tottenham Hotspur now just four points in the race to join runway leaders Manchester City in next season's Champions League.
"It's not easy to lose two games but now we are back," said Hazard. "We are Chelsea, we are the champions, we have to give everything."
At the other end of the table, a third straight defeat is another huge blow to West Brom's chances of survival with the Baggies now seven points adrift of safety and just 11 games remaining.