Rasmussen, of Denmark, completed the 106.3-mile trek from Gerardmer to Mulhouse in four hours, eight minutes and 20 seconds -- three minutes and four seconds faster than Frenchman Christophe Moreau and Voigt.
Voigt denied Armstrong his fifth consecutive yellow jersey and now leads the event by 1 minute and 50 seconds over Moreau. Armstrong is 2 minutes and 18 seconds back.
Rasmussen is fourth overall, 2 minutes and 43 seconds behind Voigt.
Alexandre Vinokourov of Kazakhstan rounds out the top 5, 3 minutes and 20 seconds back.
Following a day of rest on Monday, the race heads to the Alps on Tuesday with a 119-mile climb in Stage 10 from Grenoble to Courchevel.
Notable deaths of 2014 [PHOTOS]
UPI Sports Results for September 19, 2014