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.
Promoted Eintracht joined Bayern Munich as the only team with maximum points from their opening three games.
Hamburg, who narrowly escaped relegation last season, are one of two teams who have yet to get off the mark alongside Hoffenheim, who earlier on Sunday lost 5-3 at previously winless Freiburg.
"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."
Inui pounced on a mistake by Jeffrey Bruma to open the scoring in the 13th minute and Occean added another five minutes later.
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.
Both teams had chances to score again in a thrilling final quarter.
Eight players shared the goals in the earlier game which saw Hoffenheim take an early lead, then twice come from behind to make it 3-3 before goals in the last 10 minutes from Cedrick Makiadi and Sebastian Freis won the game for Freiburg.
