Platini, along with Sepp Blatter -- whom he hopes to succeed as FIFA president -- was suspended for 90 days
by FIFA's ethics committee last week amid an investigation into a "disloyal" $2 million payment from Blatter to the former France captain in 2011 for work allegedly carried out between 1999 and 2002.
A UEFA statement released after an emergency meeting
at its headquarters in Switzerland, Thursday said: "We support Mr Platini's right to a due process and a fair trial and to the opportunity to clear his name.
"We strongly call on all instances involved in the current process -- Fifa's ethics committee, appeals committee and CAS [the Court of Arbitration in Sport] -- to work very rapidly to assure there is a final decision on the merits of the case by mid-November 2015."
The organization's general secretary, Giovanni Infantino, told a news conference: "The main point was that we should give Mr Platini the chance to have a fair process.
"What counts is a final decision by a court. Everyone has a right to a fair trial.
"Michel Platini is a candidate for the FIFA presidency. The FIFA Congress which is scheduled to take place on Feb. 26 has to take place then. It should not be delayed.
"If someone else comes up [as a candidate], if another European comes up, it's too early to say. It was discussed, and no decision was taken."
The Russian Sports Ministry issued its own statement in which it said UEFA's executive committee had expressed "full support" for Platini.
Vitaly Mutko, the sports minister and a member of FIFA's executive committee, attended the talks in Nyon and the statement said "all members [of the executive committee] have expressed absolute trust."
That was echoed by Austrian FA President Leo Windtner, who told reporters: "We got a unified position of UEFA -- all nations give our support to Mr Platini to have fair treatment."
Both Platini, who has insisted he still intends to stand for the FIFA presidency
, and Blatter could yet face longer bans pending the outcome of the payment investigation.
Lawyers representing the Frenchman addressed the UEFA committee, which was chaired by Spanish FA president Angel Maria Villar in his absence.
Earlier this month, the Swiss attorney general's office said Blatter was suspected of making a "disloyal" payment -- against the interests of FIFA -- to Platini, who is being treated as "somewhere between a witness and an accused person" under Swiss law.
The payment, explained as being for work that had been completed nine years earlier, was received in February 2011, shortly before Blatter's re-election for a fourth term as FIFA president.
Platini argued that it had been agreed orally and claimed the long delay was "because of FIFA's financial situation at that time."
Speaking ahead of the meeting of the talks, Allan Hansen, the former head of the Danish Football Association, and David Gill, of England's FA, had indicated their countries could consider withdrawing support for Platini.
Hansen told reporters he wanted to get more information "because I didn't so far."
On Wednesday, Prince Ali bin Al-Hussein, who also hopes to succeed Blatter, warned against any delay
to the FIFA presidential elections amid the ongoing turmoil.