James Poole's stunning first-half strike was enough to separate the two sides and heap the pressure on Blades boss David Weir.
Weir's opposite number was delighted to see Pools build on Saturday's impressive 4-1 win at Mansfield in League Two.
"I thought we defended really well and counter-attacked well. I think we can hit teams on the counter-attack, especially away from home," said Cooper.
"When you're playing away from home, I think because of what we have available, we have the ability to cause problems for teams on the break.
"I think everybody has to take a pat on the back. We've come off the back of a nice victory and won away again.
"It's an 'us against the world' mentality and we have to have that.
"My feeling is that you put the strongest team you can out (in the JPT) because a win breeds confidence within the squad. I think the lads who've come in here have done themselves a huge credit."