The Dutchman, who was handed the captain's armband last weekend, scored twice in the first 20 minutes as Thorsten Fink's men picked up their second win in a row to move to within a point of the top six.
Dani Schahin scored a consolation for Fortuna, who ended the game on the attack, but were unable to find an equaliser.
Their eighth game without a win leaves them just three points above the relegation zone with four games of the season remaining.
Van der Vaart gave Hamburg the lead in the 14th minute at the second time of asking.
His first effort from Marcell Jansen's cross was saved by Fabian Giefer, but the ball stood up for him to head in from close range.
Hamburg looked particularly motivated and it took them just five more minutes to double their lead.
It was Van der Vaart again who provided a cool and cheeky finish to thread the ball through Giefer's legs after dribbling past Martin Latka and Stelios Malezas with ease.
The one-way traffic was interrupted when Schahin reduced the arrears 10 minutes before half-time with a header from close range.
Hamburg had the chance to re-establish their two-goal lead early in the second half through Heung Min Son, but he failed to make the most of that opportunity and the rest of the second half belonged to the visitors.
Latka, keen to make up for his mistake on Van der Vaart's second goal, hit the crossbar as Norbert Meier's men searched for an equaliser.
Another header from the giant defender just missed the target as Hamburg managed to hold on for all three points.