Peat called a meeting of both parties and was present at Wednesday's clear-the-air talks at Hampden Park between Hughes and the SFA's head of refereeing development, a meeting that was described by the SFA as "positive and productive".
Peat told the Herald: "The way it was heading, we would have ended up in chaos - or anarchy - if something wasn't done.
"It was not doing anybody any good, so I decided to get involved. I didn't want the meeting to be about fines or disciplinary action; I wanted to get everyone round the table and discuss it like adults because it was damaging for everyone concerned.
"It was becoming petty and tit-for-tat and I feel it was time to intervene."
Peat is looking at ways to avoid the kind of bickering between managers and officials that has marred the start to the season.
The SFA president will recommend FIFA scrap the compulsory retirement age of 47 for referees for domestic competition, which led to the loss of experienced officials such as Hugh Dallas, Kenny Clark and Willie Young.
"The meeting between managers and referees has clearly not been 100 per cent successful," said Peat.
"There are still one or two smouldering issues and we have to try and address them because the whole thing is becoming embarrassing.
"Referees are human. We all know they have made mistakes, but it should be remembered that managers and players make them as well and sometimes they can affect a result."