Norwood, 24, insists Michael O'Neill's side, who are unbeaten in 10 matches after Monday's win over Slovenia, are continually underestimated.
Northern Ireland will face Poland, Ukraine and Germany in Group C in France this summer.
"We are not going there to make the numbers up," Norwood told BBC Sport.
"The lads were saying that we have had not much recognition for what we did achieve. I think it's worked in our favour a little bit to be honest because people do underestimate us a bit and it's worked out for us.
"I think we do surprise a few teams because of how good a team we are. But we've earned the right to take part in the tournament and we know what we are good at and that we can achieve good things."
Norwood has established himself as a regular starter alongside captain Steven Davis in Northern Ireland's midfield.
But he believes that he, along with Reading team-mates Paul McShane, Chris Gunter, and Hal Robson-Kanu, must maintain their club form to ensure a seat on the plane to France.
Gunter and Robson-Kanu have respectively earned 66 and 30 caps for Wales, while McShane has played 31 times for the Republic of Ireland, both teams also Euro 2016 qualifiers.
Reading have not won any of their last five matches and are only eight points above the Championship relegation zone.
"If you are not doing well at your club and not playing then it gives the international managers a decision to make," Norwood added.
"You do not want him to be making the decisions and then not make squad for the finals. It's important to get back in the team here [at Reading] as quickly as I can and play well until the end of the season.
"We have got to start picking up points, we do not want to be getting drawn into the bottom of the table. We are only eight points off it and we know we are not safe yet."

Researchers from Aberdeen University said their find raised the possibility of bigger finds elsewhere.
The discovery is unusual as gold is normally found in "veins" deep in the Earth's crust, which are produced by water flowing at high temperature.
Gold has previously been found in red sandstone in England but not to the consistency discovered in Millport.
The Isle of Great Cumbrae in the Firth of Clyde is off the coast of North Ayrshire.
Professor John Parnell, from Aberdeen University's School of Geosciences, said the find within the rocks from Millport was surprising.
"In undertaking this research I was initially interested in analysing little structures found within common red sandstone rocks which we took from the beach at Millport, which to the naked eye appear as tiny black spots," he said.
"By analysing the structures using a high-powered microscope, we discovered that they contain concentrations of gold, which is unusual because gold normally tends to be concentrated by hot waters rising up through cracks in the Earth.
"In this case the gold has been formed and concentrated in a completely different environment, which raises the possibility of similar structures appearing within red sandstone elsewhere but on a much bigger scale."
Prof Parnell added: "While there is certainly no prospect of a modern-day gold rush in Millport, this analysis is helping us to understand unusual circumstances where gold can be formed, which in the future may even lead to discoveries that are commercially viable."
The Aberdeen University study has been published in the Journal of the Geological Society.
Reading's Oliver Norwood believes Northern Ireland will continue to fly under the radar at Euro 2016 - their first major tournament for 30 years.
[NEXT_CONCEPT]
Tiny specks of gold have been found in sandstone rocks from a beach at Millport on the Isle of Great Cumbrae.