Click here to see the winners in full
At the award ceremony at the Old Vic Theatre in London, Lord Lloyd-Webber said of the musical: "It is more contemporary and challenging than some of my earlier shows.
"Brilliant lyrics'
"I think I speak for everybody involved in the show to say that we are very proud of it and to get recognition like this is a very important part of making sure
theatre works.
"With Ben Elton's brilliant lyrics, we set out to attract a new, younger audience to musicals and happily it's working.
"I am very pleased that the theatre critics by voting it Best Musical have shown their appreciation of what Ben and I have attempted."
Despite winning an Oscar, Tonys for his Broadway shows and Grammys for his music in the US, critical success has eluded Lord Lloyd-Webber in the UK, although his musicals, which include Cats and Joseph and the Amazing Technicolour Dreamcoat, have been an undoubted success with the public.
But Lord Lloyd-Webber said he had great respect for the critics even if they did not always give him an easy ride.
"They are absolutely necessary and it is a tough job.
"At the time The Beautiful Game opened there were about 30 shows opening in as many nights so it's not easy."
Other winners in this year's awards include Joe Penhall's powerful drama about race and mental illness, Blue/Orange, which won best play and also netted a best newcomer for its star Chiwetel Ejiofor.
Best actor
Blue/Orange, originally a Royal National Theatre production which will have a new run at London's Duchess Theatre in April, and Ejiofor are also nominated for the Olivier Theatre Awards next week.
Actor Michael Gambon, best known for TV roles in The Singing Detective and Wives And Daughters, was named best actor for his role in Cressida and The Caretaker.
Victoria Hamilton, who won the Critics' Circle newcomer prize in 1995, won best actress for her performance in As You Like It.
Simon Russell Beale, who appeared in Channel 4's A Dance To The Music Of Time won the best Shakespearean performance for his Hamlet at the Royal National Theatre.
The Critics' Circle was established in 1913 and has 80 members in the Drama Section, including all of Britain's leading theatre critics.
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The winners in full
Best new play
Blue/Orange by Joe Penhall (Royal National Theatre)
Best musical
The Beautiful Game by Andrew Lloyd Webber/Ben Elton
(Cambridge Theatre).
Best actor
Michael Gambon for The Caretaker & Cressida (Comedy Theatre, Albery Theatre).
Best actress
Victoria Hamilton for As You Like It (Crucible Theatre,
Sheffield, Lyric Theatre Hammersmith).
John and Wendy Trewin Award for best Shakespearean performance Simon
Russell Beale for Hamlet (Royal National Theatre).
Best director
Michael Grandage for As You Like It/Passion Play/Merrily We
Roll Along (Crucible Theatre, Sheffield, Lyric Theatre Hammersmith, Donmar
Warehouse).
Best designer
Paul Brown for Richard II/Coriolanus/Tempest (Gainsborough
Studios, Almeida Theatre).
Most promising playwright
Joanna Laurens for The Three Birds (Gate Theatre, London).
Jack Tinker award for most promising newcomer
Chiwetel Ejiofor for
Blue/Orange (Royal National Theatre).