I saw somewhere they are the first team to do so since '45.
I saw this too, but they were saying NJ is the first team to force a game 6 after being down 3-0 in a series since 1945. The OP is just asking who was the last team to a clinch a Cup in a game in which they went into the 3rd period trailing.
This doesn't exactly fit the criteria, but it's the same concept...
1962 Stanley Cup Finals Game 6
Hawks and Leafs are tied going into the 3rd period. Hawks score about 9min into the 3rd to take a 1-0 lead. Leafs score 2 goals in the second half of the 3rd period to win 2-1 and take home the Cup