The filing included an addendum with some redactions that goes into more specifics about his a cooperation.
Mr Flynn has pleaded guilty to one count of making materially false statements to the Federal Bureau of Investigation.
By contrast, the president has been more sympathetic toward Flynn, repeatedly lamenting how his life has been destroyed by the special counsel's probe.
On Monday, Trump praised longtime associate and Republican operative Roger Stone, who has been under scrutiny by investigators over his comments about leaked Democratic emails during the 2016 presidential campaign.
Mueller said that Flynn had provided "substantial assistance" to his inquiry into allegations of complicity between the Trump campaign and the Russian government, heavily redacting specific details of Flynn's cooperation. Mueller's team told the judge it needed to redact information in the six-page memo because it contains "sensitive information about ongoing investigations".
The memo points to Mr Flynn's "exemplary" military and public service and that he "deserves credit for accepting responsibility in a timely fashion and substantially assisting the government".
Flynn, who turns 60 on Wednesday, is among a number of people in Trump's orbit who have pleaded guilty to criminal charges.
But several parts of the addendum were blacked out in the publicly available version, including a section describing a criminal investigation, a second investigation that is entirely redacted, a portion describing interactions between the Trump transition team and Russian Federation and a brief description of one way that Flynn cooperated.
The new cache of Cohen testimony might well also ensnare the president's son Donald Jr., his daughter Ivanka and her husband-in-law Jared Kushner, of whom the latter two remain as White House officials.
And this week, Mueller is expected to file a sentencing memo on Flynn.
Mr Flynn had been a fervent supporter of Mr Trump during the election campaign.
Trump was also asked if he was anxious about what else Cohen might tell investigators.
FILE - Russian Ambassador to the U.S. Sergei Kislyak arrives at the State Department in Washington, July 17, 2017.
Flynn pleaded guilty last December to lying to the Federal Bureau of Investigation about conversations about sanctions with then-Russian Ambassador to the US Sergey Kislyak. "I don't think it rises to the level of obstruction yet, but it certainly would cause people who are conducting the investigation to start asking questions about whether or not the target has reached out to them". In a series of tweets, Trump said Cohen was a "proven liar" and said he should serve a "full and complete" jail sentence. He belatedly registered with the Justice Department as a foreign agent for the work weeks after he left the White House. He did all this without first obtaining a traditional, ironclad deal under which the government would commit to seeking leniency on Cohen's behalf when he is sentenced on December 12.
Flynn said he never urged Russian Federation to vote no on a UN Security Council resolution condemning Israel's settlement policy.
Michael Flynn - who became national security adviser to Donald Trump after helping him win the 2016 presidential election - began co-operating with the inquiry led by Robert Mueller after quitting the White House past year.