The Kazakh outfit have been handed a license to compete at the top level again next year despite several failed tests in recent months.
They still face being audited by the Institute of Sport Sciences of the University of Lausanne (ISSUL) and must adhere to new operational requirements from next year.
However, Cookson has also called on those associated with the team to be extra diligent in ensuring Astana do not further tarnish the sport.
"This is not just a matter for the UCI," Cookson told Sky Sports. "We have all got our responsibilities here; the teams, Astana, the national federation of Kazakhstan, the sponsors, the funders, the suppliers of the team.
"The reputational damage that is being done to them is surely substantial, so we have all got to step up to the plate here.
"We are doing our bit, but we need other people to take their responsibility as well, so I am calling on the management of Astana to take their responsibilities very seriously."
Team Sky rider Peter Kennaugh is among those to have spoken out about the decision to grant Astana a license.
He tweeted: "What a joke this sport can be! The clean riders of the peleton need to get together and push these cheats out - enough is enough."
Cookson has sympathy with such views.
"Astana are drinking in the last-chance saloon," he added. "They've had enough assurances in the past and we're now getting to a situation where the straws are very close to breaking the camel's back.
"We need to make sure that when we take action it is something that is going to stick and be legally defenceable.
"What I really don't want to do is take action that is then subject to appeals at CAS [the Court of Arbitration for Sport] or another court and thereby set a precedent of failure. I want to make sure that we have a precedent of success."