A terrible start to the campaign with five defeats in six matches has been repaired under new coach Fink who has stretched their unbeaten sequence to seven, six under his command.
The latest success was a 2-0 victory over Nurnberg last week which left Hamburg in 11th place and with genuine hope that they can challenge for a European spot this season.
"We never want to be satisfied," Fink said this week. "We can have inner pleasure at the moment but we must always go for more success."
Fink, who joined in October from Basel replacing Michael Oenning, has quickly managed to instil a winning mentality in the only team never to have played in the second division since the introduction of the Bundesliga in 1963.
He also seems to have found a right mix for his players, saying the team was like an orchestra.
"Not everyone can be a drummer. It is important to have the right mix, like an orchestra."
Their perfect top-flight status looked anything but secure earlier this season until the arrival of the 44-year-old German.
"The team must have its sights set higher," Fink said. "Players must be proud and give it everything they have."
Bundesliga leaders Bayern Munich, who used a largely second string team in their 2-0 Champions League defeat to Manchester City on Wednesday having already reached the last 16, travel to Stuttgart to defend their one-point lead on Sunday.
Champions Borussia Dortmund, who ended their disappointing Champions League campaign with a 3-2 home defeat by Marseille on Tuesday, hope to bounce back with a home win over lowly Kaiserslautern also on Sunday.