Hi Rodrigowmier -
That is a short drive, and as long as you do not have a storm the day you wish to drive, it should be possible for a driver who is accustomed to winter driving.
Google Maps offers two routes.
www.google.com/maps/di r/23030+Livigno,+Provi nce+of+Sondrio,+Italy/ Saint+Moritz,+Switzerl firstname.lastname@example.org,9.9741 616,11z/data=!4m14!4m1 3!1m5!1m1!1s0x47836dfa 067fe897:0x5b4999f522f 8aacb!2m2!1d10.1357319 !2d46.538636!1m5!1m1!1 s0x478482076dc01a7b:0x 279fdbbd3ec97825!2m2!1 d9.8355079!2d46.490797 3!3e0
The longer one (68 minutes) goes northeast to the Swiss National Park, picks up route 28, and joins the valley of the River Inn at Zernez. Then it proceeds along the relatively easy route 27 along the valley floor until you reach St. Moritz. Maximum altitude is about 1800 meters.
I have driven the Swiss part once y0u reach Rte 28. Those roads do not not take high Alpine passes once in Switzerland, and are good roads.
The other, shorter, route (51 minutes) goes northwest through Italy to Pontresina. Google maps says it takes about three fourths the time for a route that is about 7/10ths as long....thus, a little bit slower roads. It definitely is a higher altitude route, and includes the Bernina Pass at around 2300 meters.
That altitude difference is meaningful, and I don't know if the Bernina Pass is kept open for auto traffic in the Winter.
I did a little searching and could not find the website that I usually use for European road pass conditions. My guess is that the Bernina will not be open, but you'll have to find a better information source than I am.
With the limited knowledge that I have, I'd choose the slightly longer, lower route anyway, whether the Benina route is open or not. And, I'd not travel if a storm is forecast.
Perhaps someone with more actual knowledge can jump in.
PS - I went back to Google maps, and set the "leave now" box to February 1. The route through Zernez is the only one offered. That's a clue.