The Gunners are reflecting this morning after their Champions League semifinal defeat to PSG in Paris on Wednesday night.
But Round insisted this morning on BBC Sport: "I think the fans will be really pleased with the progress. There's 18 other teams in the league that I think would swap with Arsenal right now.
"They're still second in the league, pushing all season but came up against an exceptional Liverpool. Three seasons at second, Champions League semi-finals. I think the team is growing and the silverware will come.
"I think the whole club is doing an amazing job at the moment."
Intense Arteta desperate to go next step
Round also insists Arteta remains the right manager for the Gunners.
He also said: "His focus and intensity to win and to win for Arsenal is incredible. I don’t think this will deter him one bit – of course he will be incredibly disappointed not to go to the next step.
"But you have to remember this team is in its infancy in the Champions League, it’s in its second season. You look at Man City who were in the competition for a long time before they managed to win it.
"I think you grow and you learn and you build. I would have thought there will be a huge resolve at the club to try and take the next step."