Allen scooped the 'Best Track' gong for 'The Fear', while Kasabian's West Ryder Pauper Lunatic Asylum was named 'Best Album'.
Muse were given the award for 'Best Act In The World' and U2's album The Unforgettable Fire took the 'Classic Album' prize.
Lady GaGa's 'Just Dance' was named 'Best Video' and Mr Hudson the 'Breakthrough Artist' of 2009.
Further awards were won by Edwyn Collins, Marianne Faithfull, The Specials, Yusuf Islam and Sonic Youth.
Meanwhile, Frankie Goes To Hollywood were honoured with the 'Classic Song' award for their 1984 chart-topper 'Relax'.
The ceremony was presented by comic Al Murray dressed as his Pub Landlord character. He declared 2009 a terrible year for music because of "the sad demise of Oasis".
> Click here for the full list of Q Awards 2009 winners