Defeat came three days after they suffered a record 5-0 Nottinghamshire derby loss at Mansfield Town.
Cooper is in charge until the end of the season and has been set a points target to earn an extended contract.
"I said to them 'you have got two managers the sack already and you are not getting me out'," Cooper said.
After losing his first game in charge 4-0 at Portsmouth, Cooper's Notts took eight points from a possible 12 before their defeat at Mansfield.
Asked by BBC Radio Nottingham if he saw defeat coming at 1-0 up, the former Swindon boss said: "I did because I know what is in the changing room.
"What you know is with the group of players that we have got, if any real pressure comes they will cave in."
Barnet manager and former Notts boss Martin Allen said it was a "comfortable win" against a side that "offered nothing" in the second half.
"They turned their back on their club," Allen said of the Magpies players.
"They let their supporters, their manager and everyone down from the way they played.
"I have interest in that club as I worked there, I have fond memories of my time there, and whoever brought those players in needs to take a long, hard look at themselves."
Notts, relegated from League One last season, are 17th in the League Two table after 43 games.
Notts County manager Mark Cooper told his players that they will not cost him his job after slumping to a 3-1 defeat at Barnet on Tuesday.