The home side had dominated the encounter but found themselves behind after 16 minutes to a Steven Thompson header from Kenny McLean's inswinging corner.
Conor Newton added a second in the 71st minute, just after the dismissal of Partick right-back Stephen O'Donnell for a second caution, and Thompson struck again in the final minute to put an emphatic gloss on the scoreline for St Mirren.
It was referee Beaton's decision not to award a spot-kick with the game poised at 1-0 that Archibald saw as the turning point, however, the ball seeming to clearly strike St Mirren captain Jim Goodwin's hand in the area from a Kallum Higginbotham shot.
"I think you could hear from the roar of the crowd how obvious it was that it was a penalty," said Archibald.
"Jim Goodwin moves his arm to the side and blocks the ball going towards the goal; I don't know what else is a penalty.
"The fourth official said he didn't think he'd moved his arm, but he should be embarrassed if that's what he thought because he moves his arm right to the side and it's those decisions that change games.
"It could have been one-all and we would have had the momentum to go on and get the winner, but the way things are going for us we're not getting the run of the green."
St Mirren manager and Partick Thistle Hall of Fame member Danny Lennon claimed not to have seen the incident, although he was adamant his side fully deserved the points.
"That was miles away from me," said Lennon. "Although in saying that, the 4,000-odd in the Jackie Husband stand seemed to think it was [a penalty]!
"But if you say it looks like a penalty then we've maybe rode a wee bit of luck there.
"But I thought the players thoroughly deserved the maximum points today in terms of what they brought to the game, in terms of their discipline, their desire to see that out and their patience.
"I must give Thistle a great deal of credit. They caused us a lot of problems first time around and a lot of thought went into today's game plan right from last week.
"They're a team that's well-respected, but it's about turning their strengths around into strengths for us, and I think that's what we did today."