The Dutchman, starting a second stint at the club he played at from 2005 to 2008, set up one goal for Son Heung-min but it was too late as goals from Takashi Inui, Olivier Occean and Stefan Aigner gave Eintracht the points.
Hamburg, the only ever-present side in the Bundesliga since it was founded in 1963, played the entire second half with 10 men after Czech midfielder Petr Jiracek was sent off.
"We overslept for the first 20 minutes and that is not something which should happen," said Van der Vaart, who became a cult figure on his previous spell at the club.
"After that, we saw a different Hamburg," added the Dutchman, who joined the club from Tottenham Hotspur. "There is still a long way to go."
Heiko Westermann pulled one back scored following a corner but Hamburg were reduced to 10 men on the stroke of half-time when Jiracek was sent off for a late tackle on Anderson, a decision which angered Hamburg's under-fire coach Thorsten Fink.
Aigner restored Eintracht's two-goal lead early in the second half but Hamburg refused to lie down and Van der Vaart set up their second, dispossessing Carlos Zambrano to release Son who rounded goalkeeper Kevin Trapp to score into an empty goal.
