"I had very good relations with Thorsten Fink and I hope this will stay that way but given the situation we are in we decided last night to go our separate ways," sports director Oliver Kreuzer told a news conference.
The 45-year-old Fink, who angered club bosses after leaving the team following the heavy loss to join his family in Munich, had taken over in 2011 from Swiss side FC Basel.
After a bad start to the current campaign and heavy defeats to Dortmund and Hoffenheim, Hamburg are in 15th place on four points from five games, having conceded 15 goals so far for the league's worst defence.
Hamburg, who were hoping for a Europa League spot finish before the season start, are the only team to have played in the Bundesliga every season since its creation in 1963.
He said Fink had been "surprised, shocked and disappointed" by the decision which "he did not expect."
"But I did not have 100 percent confidence in him that the team with him in charge could show the necessary power to turn it around."
Former Argentina international Cardoso had also briefly taken over as caretaker coach in 2011 before returning to the under-23 team. Former Ghana international Addo was until Tuesday Hamburg's under-19 coach.
"I am proud to have worked here for two years," Fink told northern German radio NDR earlier on Tuesday, confirming his departure. "I coached a great club. Now there is still time to reach their goals, even with a different coach."