Genia's clever running and tactical kicking kept his team going forward while his crisp passing gave flyhalf Quade Cooper the extra space to create confusion in the Highlanders' defence.
The scrumhalf's try in the 53rd minute also gave his side a 31-16 lead and enough breathing room to hold on for victory despite a Hosea Gear-inspired fightback from the home side.
The Highlanders, tipped by many to be title contenders at the start of the season, have now lost all five of their games.
The Reds dominated the breakdown in the first half with openside flanker Liam Gill superb, while number eight Jake Schatz and blindside flanker Eddie Quirk cleared the ruck out and provided the force for the visitors.
Hooker James Hanson, Gill and Schatz all scored first-half tries for the Reds, each one converted by Cooper, while Highlanders number eight Eliot Dixon barrelled over after Aaron Smith took a quick penalty.
Colin Slade converted Dixon's try and added three penalties to keep the Highlanders within sight of the Reds, who led 21-16 at the break.
Gill, who had returned after one week out following a minor knee injury, did not reappear after half time.
Genia, however, skipped over after a well-worked move from an attacking scrum when the Reds wheeled it towards the open side as Cooper darted towards the blind side, only for Schatz to feed the scrumhalf running into the gap that had developed.
Gear then almost inspired his team to battle back from a 15-point deficit when he scored a brilliant try in the 60th minute as the Highlanders counter attacked from deep.
The All Blacks winger then appeared to be everywhere on the field as he went looking for work, bumping off defenders and providing his forwards, led by replacement TJ Ioane, with some intensity and drive.
Ma'a Nonu crossed for a late try, while Lima Sopoaga, who had replaced Colin Slade converted both and added a late penalty but it was not enough for the home side to complete the comeback.