He then swept home his second from six yards after a meandering run and cross from Teale on 52 minutes.
Former Plymouth loan signing Teale drilled in a 25 yard opener off the underside of the crossbar to put County ahead in the 43rd minute.
Delighted Derby boss Clough celebrated a third win in a week ahead of next week's FA Cup clash against Manchester United and said: "It was a very good, solid away performance.
"To get the three goals was a bonus but the clean sheet was the most important thing today and that's what we asked the lads for before the game.
"It's been hard work but a good week's work for us and the players deserve an awful lot of credit for how they have gone about those three games - this last time last week for a few hours we were in the bottom three.
"Nobody likes being down there, but six points and a good cup victory as well, the players deserve an awful lot of credit for what they have done this week."
The game could have gone in a different direction had Jamie Mackie's 35-yard screamer for Argyle not come back off the crossbar before Teale's opener.
Plymouth boss Paul Sturrock said: "I didn't think there was much in it in the first half but we can't buy a win at home at the moment and their second goal, coming just seconds into the second half, took the wind right out of our sails.
"Up until then I thought it was pretty even, and that we might have gone in 1-1 or 0-0 at half time but not one down."
Clough admitted: "Teale's goal was the only thing that separated us, because I thought it was an even first half.
"Probably the 3-0 doesn't reflect the first 45 minutes, we've both hit the crossbar and had a couple of scrambles but an outstanding goal was the difference between the two teams."