Hazard waited and teed up the on-rushing Alonso, who had the presence of mind to play the ball on for Batshuayi to tuck in.
Batshuayi, enjoyed further good fortune moments later as he made it 2-0 in the stroke of half-time - Pedro and Hazard combined to find the striker, whose shot ballooned up off the leg of Jamaal Lascelles, over the goalkeeper and into the net.
Caballero then saved from Shelvey and Chancel Mbemba in quick succession before Chelsea resumed on the attack as Pedro fired just over and Alonso had a volley saved by Karl Darlow.
Alonso has carved out a reputation as one of the Premier League's finest set-piece takers and his seventh goal of the season in all competitions was a beauty, a 25-yard curler that bent out of Darlow's reach.
Chelsea FC have already been busy in the January transfer window after having signed midfielder Ross Barkley from Everton at the turn of the year.
Elsewhere, Alvaro Morata has been given time off after the death of a close friend in a vehicle accident, with Michy Batshuayi given another chance to show he still has a future at Chelsea in the Spaniard's absence.
Actor Jeremy Piven denies new sexual misconduct allegations
He added: "I have never forced myself on anyone, nor have I ever exposed myself or restrained anyone against their will". Another said she met Piven in 1996 when he was appearing on the Ellen DeGeneres sitcom " Ellen " where she was an extra.