The Belgian striker had last scored for his club in March and looked to be floundering once again when he had a header saved in an even first half.
If there had been one negative during Ronald Koeman’s impressive start as Everton boss it was the continued poor form of the club’s formerly prolific leading marksman Lukaku.
Since scoring twice in the FA Cup quarter-final against his former side Chelsea on 12 March, the 23-year-old had gone 1,139 minutes without a goal, despite 40 efforts.
Everton could have gone second in the table with a fourth goal but are tucked in just behind Chelsea on goal difference.
Sunderland remain second bottom, with David Moyes losing for the third time in as many games against the Merseysiders since he left Goodison Park for Manchester United in 2013.
Everton always looked a little more likely to score, but Jermain Defoe should have had an opener 10 minutes into the contest.
Adnan Januzaj showed great feet on the edge of the area before Defoe was fed via a deflection and he lifted a sloppy effort over the bar.
Everton had already had one attempt through Gueye but their next attack was much better, with Lukaku’s header saved well by Jordan Pickford after Bolasie’s right-wing cross.
Jermain Defoe missed a good early chance in a more even first half
Everton made a tactical switch at the break as Ross Barkley was replaced by Gerard Deulofeu and the change brought rewards as the tempo increased and they began to dictate much more.
Pickford was equal to a wicked dipping effort from Bolasie then Deulofeu and Lukaku both went close in a move which had players queuing up to tap in.
David Moyes admitted his side were poor at the Stadium of Light
Gueye’s lofted ball to the back post was perfect and Lukaku rose to nod in and end a run of 1,139 minutes without netting.
By now, Sunderland were in disarray and Everton were given far too much space in central midfield with Mirallas eventually threading through Lukaku, whose stroked finish was emphatic and drew little effort from Pickford.
Sunderland face a tough trip to Tottenham on Sunday (4:30pm), while Everton are at home against Middlesbrough a day earlier Saturday (5:30pm).