The win for the Toffees has extended their unbeaten run in the Premier League to nine games, and they now sit just four points adrift of sixth-placed Manchester United.
As for Sunderland, it was an unhappy return to Merseyside for David Moyes and they remain at the foot of the standings, three points adrift of safety.
Moments later, Pickford was not required as Lookman lashed the ball high over the crossbar from distance, but the young English stopper had to be alert to deny Gueye after he had burst into the penalty area and fired the ball towards the near post.
Sunderland were threatening sporadically but as the game approached half time, they had done enough to quieten the Everton support and would have been hopeful of reaching the break on level terms.
However, the deadlock was broken on 40 minutes when Seamus Coleman's pass picked out the run of Gueye, and he made no mistake in firing the ball past Lookman from inside the area.
Sunderland needed a response after the restart and they managed one as they earned a number of corners during the opening stages of the second half.
However, while they improved during the third quarter of the match, they were unable to create a clear-cut chance and Everton were eventually able to turn the tables.
Lukaku had endured a frustrating afternoon but in the space of a minute, he twice tested Pickford who continued to impress on his first appearance since Boxing Day.
Sunderland came within inches of a late equaliser with 12 minutes left when Jermain Defoe struck the underside of the crossbar, but his miss led to Everton doubling their lead.
Lukaku and Bryan Oviedo were left to fight for the ball on the break and after Lukaku managed to outpace his former teammate, he directed the ball past Pickford via the aid of a deflection from the left-back.
During the closing stages, Enner Valencia saw a shot brilliantly saved by Pickford but Everton were forced to settle for the two goals as they continued their run as the form team in the division.