Friday’s goalless draw against England at Wembley kept the side well and truly in the hunt for a first spell in the knockout stages of a major tournament.
And results elsewhere on Monday ensured Scotland will definitely qualify if they beat Croatia at Hampden Park.
Victory would put Scotland on four points, sealing progress to the last 16 at least as one of the four best third-placed teams.
They could even finish second ahead of England if the Czech Republic win at Wembley and Scotland turn around their three-goal disadvantage against the Auld Enemy.
Scotland will need to deal with the expectations as they bid to make history and improve on their display in their opening 2-0 defeat by the Czechs.
But Cooper, 29, feels they blew away any signs of nerves and answered doubts about their quality with their performance against England.
“There was always going to be nerves but they soon go,” the Leeds centre-back said. “You realise the job at hand and you get on with it.
“The lads who played the other night proved themselves on the world stage, that we belong here.
“The nerves have gone and it’s all focus on Tuesday night.
“It’s a massive game, probably the biggest game Scotland has seen for a very long time, and it’s one we are all very excited about and looking forward to.”
Steve Clarke’s side will need to break their Euro 2020 duck and find the net at least once - they are the only team who have not scored yet.
Cooper said: “We have created a lot of chances, against both the Czech Republic and England. We have had 30 shots.
“We just need someone to step up and put the ball in the back of the net and I’m sure we will do that.
“The strikers have confidence in themselves and if they keep getting the chances one is going to go in. We need it more than ever.”