A Stuart Noble own goal and a Lawrence Shankland strike gave the visitors the points at Balmoor as they continued their stunning recent form.
Boss Speirs said: "It was a great performance and a great victory.
"I don't think Queen's Park have won here for a number of years, certainly not in my time at the club.
"Over the years we have had a good first half or a good second half, but [on Saturday] we were excellent from start to finish.
"There was a real brightness about our play, especially in the forward areas, while our defenders stood up to the challenge.
"I was delighted with all of my players. I thought individually and as a team they played really well.
"The most pleasing thing was that we were without five or six players, yet the ones who came in looked like they have been playing every week. I cannot pay them a higher compliment than that."
At the other end of the table, Clyde dropped to bottom spot when they went down 2-0 against fellow-strugglers Stirling Albion.
Bully Wee gaffer Jim Duffy said: "When you make two basic errors you are going to be up against it."