Higginbotham fired a goal of the season contender to put Thistle ahead with just five minutes remaining, but Accies struck back immediately through Tony Andreu, and fellow substitute Jason Scotland sealed a remarkable 2-1 win in the final minute.
"We didn't pass the ball at all well," said Archibald. "But to score a wonder goal like that with five minutes to go I thought we had it.
"We played into their hands. We were the home side and I thought we should have taken the game by the scruff of the neck. We didn't do enough.
"I thought we'd learned over the pre-season and the first couple of games to be streetwise, but to lose two goals like that was criminal, and the way we lost the goals as well because I think we had enough chances to stop them. Especially the first goal, the boy's ran too far with the ball.
"But when you've just scored, that can be your most vulnerable time."
There were no such worries for Hamilton manager Alex Neil, who has now witnessed his side pick up three wins in a row after a disappointing showing in the first game of the season against Inverness.
He now believes that his players feel they belong at this level, and is hoping that the momentum they have gained through the subsequent victories over St Mirren, St Johnstone and Partick will breed more success as the season goes on.
"Performance levels are huge," said Neil.
"If you play well and they come off knowing they've played well, and they want to try and dominate the ball, work hard for each other and they're organised, but not only that, if you win games and get a bit of success, get a pat on the back and get people in the media writing good things about you, the players will take huge satisfaction from that.
"To be fair, for the past year all it's been is positive things about them and rightly so. They showed again today that they deserve all the plaudits they're getting."
Neil insists he never thought that his team were beaten, even after falling behind so late in the game. He also praised the impact of his match-winning substitutes Scotland and Andreu, who provided the composure his side had been lacking earlier in the match as they squandered a host of opportunities.
"I know what we're capable of doing and in the second half I thought we were the team with the momentum until they scored," he said.
"When we got the goal back you just got the sense that they had the bit between their teeth again and Jason was really calm and cool going in on the goalkeeper and slides it in the corner.
"The substitutes have got to take credit. It's easy to come on and get caught cold and maybe not get up to the speed of the game, but the two of them came on and they're quality professionals, Jason's 36 I think and Tony was magnificent for us last season as well. The two of them come on and have had an impact on the game and scored the two goals, so the credit has to go to them."