Look at the runs allowed in the three-game LDS sweep: 2, 3, 1, for a total of six. That is the reason the Yankees swept the Twins (and the reason they have failed to advance in the recent past).
For instance, Cleveland scored twice that in game one of the 2007 LDS. They also scored six in their clinching game four.
In 2006, Yankee pitchers gave up six and eight runs (respectively) in games three and four, losing both.
In 2005, the Yanks allowed 11 runs in game three and five runs each in games two and five (all losses).
So while A-Rod and Co. certainly got some big hits, they were big because the pitchers kept the games close. What looks like clutch hitting is largely about good pitching. The Yankees haven't so much lacked clutch hitters in recent years, but great pitchers. Like A-Rod said, the series was about three guys: CC Sabathia, A.J. Burnett and Andy Pettitte.