Calgary drafted Gillies 75th overall in the third round of the 2012 NHL entry draft.
Gillies, 21, made 49 saves for Providence in a 4-3 win over Boston University on Sunday to cap his college career with an NCAA championship.
The six-foot-six, 225-pound goalie from South Portland, Maine, was named the most valuable player of the Frozen Four tournament.
Gillies joined the Flames for a pre-game skate in Vancouver on Wednesday prior to Game 1 of their first-round playoff series against the Canucks.
He posted a goals-against average of 2.01 and a save percentage of .