White started superbly, including a 130 break in the fourth frame to take a 3-1 lead, and looked ready to reach the final when set at the table in the next one, but a miss-hit swerve allowed Hendry to clear up to reduce the arrears to one.
The Whirlwind showed no nerves in the sixth frame though and takes his place in the final.
Hendry won four World Championship finals against White at the Crucible - 1990, 1992, 1993 and 1994.
White will meet Ken Doherty in the final after the 1997 world champion beat fellow Dubliner Michael Judge 4-1 in the other semi-final.
'That was Ronnie O'Sullivan at his absolute best' - Jimmy White analysis of Rocket's amazing display