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Rodriguez was victorious on the mountaintop finish to the 143km stage, finishing four seconds ahead of Froome, of Sky, and 22 seconds in front of Australia's Cadel Evans of the BMC team.
Rodriguez's compatriot Alberto Contador was fourth, 27 seconds back.
In the overall standings, Froome took the red jersey from Slovakia's Peter Sagan and now leads by a margin of 24 seconds from Evans and 25 seconds from Contador.
Sagan was some way off the pace on the day, as was Froome's compatriot, and Sky team-mate, Bradley Wiggins.
Friday's fifth stage is a 144km ride from Al Alam Palace to the Ministry of Housing in Boshar.
The six-stage race concludes on Saturday.