The Pools boss saw his side record a seventh win in their last eight games to reach the second round of the FA Cup.
"This is the third team we have played from League One this season in cup competitions and we have been better than them all," said Cooper after the 3-2 win.
"We had to defend at the end, but that was Notts throwing the ball over us as they couldn't pass through us.
"We made the players aware of what a new manager would give them - energy and enthusiasm.
"I have to say Notts County did try and play nice football, but we were better than them at playing football and scored three lovely goals. Luke scored from two rebounds.
"I'm pleased with the effort, application and the way we played football. I reiterate we were better than them.
"We were the better team by a long shot. They resorted to knocking long balls aiming for a knock down - it shows what teams have to do to beat us right now."
Hartlepool took the lead after 16 minutes, Jack Baldwin meeting Simon Walton's free-kick to head home.
But Notts were level less than two minutes later. Scorer Dean Leacock did not know much about it as Matty Dolan's clearance crashed off the County captain and bounced into the back of the net.
Luke James put Pools ahead again three minutes before half-time, reacting quickest after goalkeeper Bart Bialkowski spilled Michael Duckworth's shot.
In a tame second-half, Pools made it 3-1 after 75 minutes. James was quick off the mark again to net his eighth goal of the season after Dolan's free-kick hit a post.
Ronan Murray crashed in a shot from close range with six minutes remaining to get the visitors back within a goal but Derry's team could not force another equaliser to get a replay.