Train from Varenna to St Moritz

Answer: Yes.

It's not all necessarily served by the "official" Glacier Express trains with panorama cars, but the view is good from any train. Travel time is 4 hours each way. You could catch a panorama train on the morning leg, like this:

Varenna-Esino dep 09:24
 
R 2554 Regionalzug Direction: Tirano

Tirano arr 10:52
 
Transfer time 8 min.

Tirano dep 11:00

R 1644 Regionalzug Direction: St. Moritz, Panorama coach (Reservation compulsory)

St. Moritz arr 13:11

See return times and other options through the DB link at Looking Up Train Schedules and Routes Online, or SBB.ch or Trenitalia.com.

Very nice day trip. The transfer at Tirano is to a different train station, but they are across the piazza from each other. 8 minutes is tight but doable. You might want to take the train from Varenna one hour sooner at 8:24 because they are 2 totally separate tickets and if the train from Varenna is just a little bit late, it could get hairy. You can walk around the town for a while waiting for the Bernina Express. There is no ticket facilities at Varenna-Esino station, so buy in advance at the travel agency or tabac in Varenna beforehand. You can buy Bernina Express tickets at the Rhb station in Tirano and there is a small supplement for the Bernina Express train with panorama cars.