Chelsea forward Hazard opened the scoring at the Spartak Stadium in Moscow with an early penalty.
Manchester United striker Lukaku struck either side of Tunisia defender Dylan Bronn's header.
Chelsea striker Batshuayi added a fifth late on, ahead of an added-time consolation from Wahbi Khazri, the Sunderland winger.
Their next opponents are England on Thursday 28 June, when Tunisia face Panama.
Mexico qualified for the knockout stage with the help of a Chicharito strike.
Spurs' Son Heung-min was on target in added time for the Asian side.
Mexico next meet Sweden on 27 June, when South Korea face Germany.
Man Utd's Victor Lindelof must have thought his side were on course for victory when Ola Toivonen gave them the lead.
But Marco Reus' effort and Toni Kroos' stoppage-time free-kick kept German hopes of progressing alive.