Henman through in London, Rusedski falls
The Swede was previously unbeaten at the tournament at the Royal Albert Hall, having overcome Jeremy Bates and Henri Leconte, but he found his match in Henman.
The former British number one raced to a three-game lead and was able to maintain it to take the first set 6-3.
Enqvist, ranked number one on the ATP Champions Tour, rallied to take the second set and force a champions tie-break but Henman, roared on by the home crowd, took a two-point lead at 4-2 to eventually claim victory 10-7.
Henman goes on to face Todd Martin in the semi-finals after the American beat Cedric Pioline.
The Frenchman had looked to be sweeping to a victory over until an injury derailed his match.
Pioline had taken the first set 6-3, but had to retire at 1-2 down in the second set with a calf problem.
There was very little to separate the two left-handers, both famed for their booming serves.
Ivanisevic will play Pat Rafter in the final four in a rematch of the 2001 Wimbledon final, which the Croat won to claim his only career Grand Slam.
Rafter won his place in the semis with a 6-1 6-7 (4) 10-3 win over Henri Leconte in the final match of the day.