The "E! Fashion Police" co-host, 28, and Mosshart, 29, met at model Kate Moss' wedding in July 2011.
Osbourne and Mosshart got engaged a few weeks before Christmas, UsMagazine.com said Wednesday.
"Kelly is the happiest she's ever been," one of her friends told the magazine. "Her career is going well, and she's in a healthy relationship with a wonderful guy."
2014: The Year in Fashion [PHOTOS]