Ally Samatta struck early in the second half to earn the hosts a draw, however - a fair reflection on the basis of the 90 minutes, which faded out as the game wore on.
Belgium Under-21s international Trossard was at it again moments later, this time turning on the edge of the area and flashing the ball narrowly wide of the target.
Both sides were seeing plenty of the ball but struggling to create too many opportunities in the opening 40 minutes or so, with Rooney's deft flick forcing a rare save out of Nordin Jackers.
Trossard was unlucky not to open the scoring at the third time of asking when shooting narrowly wide at the end of a good passing move, while Rooney again came close with a 30-yard free kick, picking out the roof of the net on this occasion.
Just when the Belgian side were growing more and more into the match, testing Pickford twice in quick succession through Benson Manuel and the lively Trossard, Rooney appeared at the back post to help home Sandro's killer cross from the right.
Genk were level within 10 minutes of the second-half restart, though, courtesy of Samatta's strike down the middle after substitute Cuco Martina only half-cleared a cross into the box.
Everton had Maarten Stekelenburg to thank for not falling behind two minutes later, with the replacement keeper somehow keeping out Samatta's diving header by helping the ball onto the crossbar.
Everton now turn their attention to next Thursday's Europa League tie against Slovakian outfit Ruzomberok, officially marking the start of their 2017-18 campaign.