Guillaume Hoarau gave the hosts the lead in the 10th minute but Everton replied in emphatic fashion as Lukaku scored either side of Seamus Coleman’s strike to make it 3-1 at half-time.
It got even better for Everton shortly before the hour mark as Lukaku completed his hat-trick.
John Stones was sent off minutes later for bringing down Hoarau in the box but the striker fired his penalty over the bar.
Roberto Martinez’s side had only won two of their last 16 matches in all competitions but another fine performance on on the continent puts them in pole position to reach the last 16, ahead of the return leg on January 26.
The Swiss club were expected to be a tough challenge on their artificial pitch after winning all five of their previous home games in the competition, including a 2-0 victory over Napoli.
And, after Lukaku’s early header was well saved by Yvon Mvogo and Bryan Ovideo’s mis-hit cross struck the outside of the post, it was the hosts who took the lead.
Raphael Nuzzolo cut the ball back and Hoarau fired it past Tim Howard from the edge of the box.
Everton responded in impressive style as Lukaku and Ross Barkley were denied by decent saves from Mvogo and the pressure told in the 24th minute when Lukaku headed in Gareth Barry’s cross from the left.
Within four minutes, Everton had another as Steven Naismith’s delivery across the face of goal was turned in at the far post by Coleman.
While there was a hint of offside about the goal, Young Boys responded well and Howard pulled off a superb double save to keep out Hoarau’s volley, before smothering a follow-up effort from Nuzzolo.
Young Boys again cried offside as Oviedo backheeled the ball through for Barry and he crossed for Lukaku, who was beyond the last man, to score at the back post.
A frantic first half ended with Sekou Sanogo seeing a shot blocked by a combination of James McCarthy on the line and the post.
It did not take long for things to get warmed up again after the break as Lukaku burst forward and lifted a clinical finish over Mvogo in the 58th minute to become the fourth Everton player to score a hat-trick in Europe.
Young Boys swiftly rallied but though they could have reduced the deficit after Stones – who will miss the decider at Goodison – saw red, Hoarau was off target.
Lukaku missed two brilliant chances to extend Everton’s advantage at the other end but the Toffees comfortably held on to their handsome lead, despite being a man short, to ensure they are strong favourites to make the last 16.