Ex-Labour press chief Alastair Campbell and MP Simon Hughes have also settled with News Group Newspapers (NGN), which published the News of the World.
Mr Gascoigne received £68,000 while Mr Coogan got £40,000.
The comedian said afterwards: "This has never been about the money."
Tony Blair's former spokesman Mr Campbell, who won substantial damages plus costs, also said his action had "not and never has been, about the money".
Liberal Democrat deputy leader Mr Hughes received £45,000 plus costs while football agent Phil Hughes, a friend and former agent of ex-footballer George Best, received substantial damages plus costs.
Hugh Tomlinson QC told the court his client, singer Charlotte Church, had not reached an agreement. The case involving her and her parents is due to start at the end of the month.
By Clive ColemanBBC News legal correspondent
This latest round of settlements represents a continued waving of the white flag by News International in the civil cases for breach of privacy arising out of the phone hacking at the former News of the World.
The vast majority of the original 60-odd cases have now been settled. And the more they settle, the more difficult it becomes for News International to defend one.
The case brought by the singer Charlotte Church and her family is due for trial, as are five other cases.  Whether any of them will actually get there seems doubtful.
The settlement sums reflect the level of hacking, the period over which it took place and the number of articles that were written based upon the information hacked.
But this is not even the beginning of the end. Waiting in the wings, are around 800 additional potential claimants, perhaps more. News International seems to be in for a very long and very costly act of surrender.
Speaking outside the High Court, Mr Coogan said he had taken the action to expose the way in which parts of the press had been operating.
"Like other people who sued, I was determined to do my part to show the depths to which the press can sink in pursuit of private information.
"The police and the Leveson Inquiry will be investigating these matters but at the time, when these civil cases began, News International seemed likely to succeed in covering up the hacking scandal completely.
"Neither the police nor the government were willing to hold those responsible accountable for unlawful acts."
He said some of the phone hacking victims were "many ordinary members of the public, sometimes vulnerable people with the most tenuous connection to news".
Mr Campbell said he would be using the settlement to make donations to organisations, including the Labour Party and Mind, "so that at least some small good for the causes I believe in can come out of the criminality and cultural depravity of others".
"For me, this has been about people with a voice and a platform using them to change the media culture which, as I argued at the Leveson Inquiry, has become putrid in parts," he said.
The announcement of the latest settlements, at a hearing in London, came during the latest in a number of pre-trial reviews.
Mr Campbell said his settlement was a "satisfactory outcome" and NGN had agreed to search for other documents and that he may be entitled to further damages.
Details of the effect hacking had on Mr Gascoigne were given in a statement from his solicitor.
The judge was told the former England international footballer, who also played for Newcastle United and Tottenham Hotspur, had been the subject of a number of News of the World articles over many years and had "considerable concerns" about the source of some of the "intrusive and private" information.
The statement said: "Mr Gascoigne was worried that the information was being obtained by bugging or tapping his telephone conversations, as a result of which he was accused of being paranoid.
"In addition, Mr Gascoigne was worried that the information was being given to the News of the World by his friends or family, as a result of which he fell out with several of his friends and family."
The statement said the ex-footballer apologised for wrongly accusing those close to him of leaking information.
Those receiving settlements include:
Liberal Democrat deputy leader Mr Hughes said the practice of hacking was "criminal behaviour on an industrial scale."
He added: "We must now make sure that nothing like this can ever happen again."
These latest developments mean 15 more phone hacking cases have now been settled, including nine of the 10 which were due to go to civil trial on Monday. There were 37 other claims settled with NGN last month.
The announcement, at a hearing in London, came during the latest in a number of pre-trial reviews.
Another five phone-hacking cases are still due to be heard.
Those are of model Elle Macpherson's former adviser Mary Ellen Field, footballer Ryan Giggs, former royal butler Paul Burrell, police officer Jacqui Hames and her husband David Cook, and PR consultant Nicola Phillips.
Comedian Steve Coogan and ex-footballer Paul Gascoigne are among the latest people to have settled claims for damages over phone-hacking, the High Court has heard.