Super League debutant Craig Hall missed a late conversion as Hull fell a whisker short of pulling off a stirring second-half fightback.
Quins raced into a 30-12 lead in the second half but last year's Grand Finalists would not take no for an answer and would have had a share of the spoils but for Hall's late miss.
Quins went into the game without a win at home all season and they went behind early on after Hull cashed in on a high tackle from Louie McCarthy-Scarsbrook.
The Black and Whites kicked for touch and Sid Domic spotted a gap and stepped on the gas to go over in the corner, with Hall missing the conversion.
Quins hit back and were awarded a penalty for offside, with Jon Wells on the end of a smart move to pull the scores level at 4-4 - Henry Paul also missing.
Paul made amends minutes later when he broke through the Hull line and was dragged down just shy of the 20-metre mark - with Daniel Heckenberg going over and Paul converting to make it 10-4.
Hull surged forward again and less than three minutes later Danny Washbrook evaded a tackle and passed to Kirk Yeaman, who went over in the corner - with Hall again missing the conversion.
Quins went up a gear with the half-time hooter approaching. Paul stretched the lead with a penalty and Andy Smith capitalised on a sustained period of pressure to cross, with Paul adding the conversion to leave the Black and Whites 18-8 down at the break.
Julien Rinaldi scored a solo try, with Paul converting it to make it 24-8 - but Hull's Richard Horne went over after good work from brother Graeme to make it 24-12.
A converted Mark McLinden try made it 30-12 but Horne's second reduced the lead once more.
Hall then scored a try of his own to make it 30-24 before Matt Sing brought Hull within two points with six minutes to go.