Just played a game with the Twins (me)/Tigers and hit a HR with Morneau followed by another with Young a couple of batters later. Both times I didnt see any cutscenes until they showed the replay of it. The cutscenes are awesome though, when I hit the HR with Morneau, he did stood there for like 2 seconds admiring it before starting to round the base while Scherzer (who gave up the HR) just knelt down and put his hands over his face as soon as the bat hit the ball. Both HRs were shown multiple times (like 3) before they moved on to the next batter. It was awesome.
When you first start up the game, they ask you if you want to turn on autosave and living rosters. So I assume you can turn it off in the options menu.