The song, which has been used in adverts for Zane Lowe's Radio 1 show, has been downloaded more than 31,000 times on all internet sites, a spokesman for HMV said.
Crazy, which is not available in the shops until Monday, toppled last week's top song So Sick, by Ne-Yo, from the number one spot.
Gennaro Castaldo, spokesman for HMV, said if physical copies fly off the shelves at the same rate it could become the year's biggest seller.
"It is an outstanding single so I don't think we are going to see this happening with a song every week," he added.
The duo behind Gnarls Barkley are American producer Danger Mouse (real name Brian Burton) and hip-hop artist Cee-Lo and in 2004 Danger Mouse remixed the Beatles' White Album and rapper Jay Z's Black Album to produce The Grey Album.
Gnarls Barkley will be appearing at the 02 Wireless Festival in June in Leeds and London.
