Henman, who won just his second tour match of the year on Tuesday when he beat American Michael Russell, squandered three set points in the first set tiebreak against the powerful Tursunov but battled back to win the second set.
The 32-year-old could not stay with the second seed in the decider, however, as Tursunov, who reached the semi-final at Queen's Club last week, powered through.
Croatia's Ivo Karlovic, another dangerman on grass, also progressed at the Wimbledon warm-up event when he ousted Frenchman Gilles Simon 7-6, 6-7, 6-4. Karlovic fired down 30 aces during the match.
French eighth seed Arnaud Clement continued his good form on grass when he beat Russia's Teimuraz Gabashvili 7-6, 3-6, 6-3.
