The home side made a lively start to the game but went behind to a Steven Thompson header after 16 minutes.
Stephen O'Donnell got red-carded for the home side in the second half for his second bookable offence, before Conor Newton and Thompson's second of the game saw St Mirren register only their third win of the season.
But Archibald believes the game's turning point was referee Beaton's decision not to award Partick a penalty after Kallum Higginbotham's shot appeared to strike St Mirren captain Jim Goodwin's hand in the area.
"I think you could hear from the roar of the crowd how obvious it was that it was a penalty," Archibald said.
"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 Danny Lennon claimed not to have seen the incident despite being in the next dugout along from Archibald, but felt his side were good value for the three points.
"That was miles away from me," Lennon said. "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'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."