If you have the same version of Julia installed on the remote machine (and in the same location), you could use the Distributed library to do this — just r = addprocs(["remote.server"]) and then @spawnat r[1] run(...).
@jw3126 - Did you ever find a clean solution to running remote commands over ssh from within Julia (without a Julia dependency for the remote server)? Thanks.
No, but those Julia abstractions are a subset of the functionality provided by most shells, so you could conceivably write a package that creates the proper Cmd from your Julia function calls.