The club will still have to pay a fine, which has been reduced from €30 million (£25m/$33m) to €10m (£9m/$11m).
City were initially punished by UEFA in February, when they were banned from competing in the Champions League or Europa League for two years and given a €30m fine for breaching Financial Fair Play rules.
In its initial statement confirming City's punishment, UEFA said that it found the Premier League club guilty of "overstating its sponsorship revenue in its accounts and in the break-even information submitted to UEFA between 2012 and 2016."
A statement from the CAS said City did not overstate their sponsorship contributions but did fail to cooperate with UEFA authorities.
"Manchester City FC did not disguise equity funding as sponsorship contributions but did fail to cooperate with the UEFA authorities," read the heading to the CAS statement.
"The CAS award emphasized that most of the alleged breaches reported by the Adjudicatory Chamber of the CFCB were either not established or time-barred," added CAS, which will publish its full written reasons over the coming days.
"As the charges with respect to any dishonest concealment of equity funding were clearly more significant violations than obstructing the CFCB's investigations, it was not appropriate to impose a ban on participating in UEFA's club competitions for MCFC's failure to cooperate with the CFCB's investigations alone."
City responded to the verdict with a statement which read: "Whilst Manchester City and its legal advisors are yet to review the full ruling by the Court of Arbitration for Sport, the club welcomes the implications of today’s ruling as a validation of the club’s position and the body of evidence that it was able to present.
"The club wishes to thank the panel members for their diligence and the due process that they administered."
The ruling also has an impact on the clubs currently below City in the Premier League table.
If the ban had been upheld, the team finishing in fifth place - currently occupied by Manchester United - would have qualified for the Champions League, with Europa League qualification extended to sixth and seventh.
Questions will now be asked over the long-term viability of UEFA's FFP model after the outcome of what could come to be viewed as a landmark ruling.
However, the governing body said it remained "committed to its principles" in a statement reacting to the verdict.
"UEFA notes that the CAS panel found that there was insufficient conclusive evidence to uphold all of the CFCB’s conclusions in this specific case and that many of the alleged breaches were time-barred due to the five-year time period foreseen in the UEFA regulations," a statement read.
"Over the last few years, Financial Fair Play has played a significant role in protecting clubs and helping them become financially sustainable and UEFA and ECA remain committed to its principles."
UEFA's Financial Fair Play regulations aim to maintain the financial health of European football clubs in order to avoid ruination of teams. This is effectively achieved by limiting the net losses a club is allowed to make over a specified time period.
City immediately denied the Club Financial Control Body (CFCB)'s findings and initiated an appeal against the decision at the first opportunity, with club CEO Ferran Soriano emphatically declaring: "The allegations are not true. They are simply not true."
UEFA's investigation into the matter was prompted by the publication of a number of allegations by German magazine Der Spiegel, which drew on documents said to have been obtained by whistleblowers Football Leaks, in 2018.
Animosity has been bubbling between the Premier League giants and European football's governing body for almost a decade now, and the latest clash has further strained the relationship.
Fans of City have made a habit of booing UEFA's anthem when it is played ahead of Champions League games, something that has intensified since 2014, when the club first fell foul of Financial Fair Play.