Rasmussen finished two minutes and 47 seconds ahead of Spanish cyclist Iban Mayo in Sunday's 102-mile stage, allowing him to take the yellow jersey away from German competitor Linus Gerdemann.
Gerdemann now trails Rasmussen by 43 seconds overall, while Mayo remains two minutes and 39 seconds behind the top spot in third.
Spaniard Alejandro Valverde rounded out the top three finishers in the latest leg from Le Grand-Bornand to Tignes.
U.S. cyclist Levi Leipheimer took 12th in the eighth stage, keeping the Discovery Channel Pro Cycling Team competitor 12th in the overall standings.
Swim Week Miami Beach 2014 [PHOTOS]