Finke left the club "by mutual consent" after Saturday's 1-0 win over Hertha Berlin with Dr Wolf insisting the move was made in the club's best interests.
"It is all about football and it is all about the success of FC Cologne," he said in a statement.
"Last night, we spoke with Volker Finke after our home game against Hertha Berlin and, afterwards, we announced that he and Cologne would be parting company by mutual consent.
"The reason for this is a difference in opinion in view of the laying of foundations for the footballing future of Cologne.
"I would like to express my thanks for Volker Finke's work over the past year. He made a valuable contribution to us avoiding relegation in the 2010/11 season and, during his time with Cologne, he has given the club new impulsion."
Finke arrived at Cologne last year and, shortly after taking over as the club's director of sport, he found himself on the bench as coach after Frank Schafer handed in his resignation.
He took charge for the final three games of the season, winning all three to save the club from the drop before reverting to his role of director of sport in the summer and appointing Stale Solbakken as the club's new coach.
However, his relationship with the Norwegian tactician has been frosty in recent weeks, leaving a situation in which one of the pair had to leave, and Finke has drawn the short straw.
"We feel that we need clarity over the future in our current situation because Cologne needs the coach and the team to have full concentration on the remaining games," explained Wolf.
"It had become clear to us that, in some fundamental aspects of our work and the direction we wanted to take, there were differences of opinion (with Finke).
"Therefore, we came to the common conclusion that it made no sense carrying on working together."
Solbakken had not seen eye-to-eye with Finke after hinting that he had overstepped his responsibilities and was interfering too much with his own field of duty.
Finke signed North Korea international Jong Tae-Se during the winter break without even consulting Solbakken until the deal was done, and the pair's relationship has deteriorated since then.
The Norwegian coach will now take sole responsibility for team affairs after receiving his club's support.
"Stale Solbakken enjoys our full backing, as do our current team," said Dr Wolf.
"We are convinced that we have the quality to avoid relegation and achieve our goals beyond that. The management team will analyse the current sporting situation with Stale Solbakken.
"We are keen for yesterday's [Saturday's] win to lead into a sustainable run of positive results."