Thistle entered the game still seeking their first home league win of the season but Conrad Balatoni and Lyle Taylor (2) struck in the second half to punish Aberdeen's Barry Robson for his first-half penalty miss.
"I'm absolutely delighted," said Archibald.
"We showed great character in the second half, after a first half when we appeared nervous and were not passing the ball well.
"We've got to go on from this and continue to show more belief and character. It's a big lift for us that we have also moved up a place in the league table."
Partick had given away a two-goal lead at Motherwell last week and Archibald admitted: "It was hard to celebrate after the goals, as you don't know what's round the corner.
"But I'm delighted with the win. We worked all week with Lyle turning and running at their centre-backs, and it paid off today."
Dons boss Derek McInnes said: "Credit to Thistle. They deserved the win and were much more clever than us in the last half-hour.
"The players were desperate to get right back in it. They are used to winning, but we have no divine right to win all the time."
However. McInnes did admit to feeling a little aggrieved about one of referee Alan Muir's decisions.
"For me there were two pivotal moments in the first half," he said.
"Obviously, things would have been different if Barry had scored his penalty.
"But then, after that, it was a clumsy challenge on Adam Rooney, and he was the last man so I thought he should have been sent off.
"But we have to make amends, starting with the match against Celtic on Tuesday."