The 19 year-old, son of ex-Manchester City and Leeds United defender Alf Inge Haaland, became only the eighth player to score three times on debut in Europe's elite competition.
Hwang Hee-chan, Dominik Szoboszlai and Andreas Ulmer were also on target for the Austrian champions as they moved top of Group E.
Norway international Haaland, who had already scored 11 times domestically heading into the match, took just two minutes to open his Champions League account when he fired low into the left corner from inside the box.
He soon doubled their advantage when a quickfire counter-attack ended with the teenager being sent clean through and he made no mistake, tucking confidently inside the right post.
A poor Genk clearance was then punished as Hwang steered a shot low inside the right post after being fed by Zlatko Junuzovic.
Genk gave itself a glimmer of hope five minutes before the interval courtesy of a sliding finish from Colombia defender Jhon Lucumi after Salzburg had failed to deal with a set-piece.
However, two quickfire Salzburg goals - the first a close-range finish from Haaland to complete his hat-trick before Dominik Szoboszlai converted Takumi Minamino's cross - ensured the contest was over before the break.
Haaland became the first teenager to score more than once on his Champions League debut since Wayne Rooney's hat-trick for Manchester United against Fenerbahçe in September 2004.
Mbwana Samatta's header pulled one back for the visitors, yet Ulmer restored the four-goal advantage midway through the second half with a tidy finish after good work again from Junuzovic.
Samatta was initially shown a red card for a challenge on Junuzovic only for the decision to be overturned by VAR.