Hamburg, who returned to the Bundesliga after seven seasons in the second division this campaign, offically beat the drop last week with a big win over Eintracht Frankfurt.
The 1983 European Cup winners were in fine form again to defeat Freiburg, who scored two goals through Igor Matanovic.
Freiburg could have secured a European spot with victory but will have a chance to reach the Champions League and win a first major trophy in the club's history if they beat Aston Villa in Istanbul on May 20.
With 14 minutes gone, Jatta opened the scoring with his first Bundesliga goal after decade at the club, but Matanovic struck back two minutes later.
Tottenham loanee Vuskovic, 19, returned to the starting XI for the first time since early April, put Hamburg ahead again with his sixth goal of the season, before Balde added another.
Matanovic scored with three minutes remaining but Hamburg held on for a ninth win of the season.
Later on Sunday, rock bottom Heidenheim can keep their slim hopes of beating the drop alive with a win away at Cologne, while Mainz host Union Berlin.