The 36-year-old former prison officer sold twice as many copies of her single, Skyscraper, than US star Pharrell Williams in second.
Bailey, 36, who was crowned the winner of the singing contest last week, is splitting all proceeds between Great Ormond Street Hospital Children's Charity and Together For Short Lives, a charity that helps children with life-threatening conditions.
Chancellor George Osborne has agreed to waive VAT on the release.
Bailey said: "Being No 1 is absolutely amazing. It really is the icing on the cake and it means Great Ormond Street Children's Charity and Together For Short Lives will benefit, which makes it extra special."
Bailey sold 149,000 copies of Skyscraper, outstripping Happy by Williams by more than two copies to one to become one of the fastest selling singles of the year.
One More Sleep, by the 2006 X Factor winner Leona Lewis, took the third spot.
Robbie Williams held on to top spot in the album chart with Swings Both Ways.
His record was neck- and-neck midweek with One Direction's Midnight Memories, but beat it to to the top spot by selling just 200 more copies.