Spurs have not qualified for the lucrative competition due to Chelsea's triumph in the final against Bayern Munich and will enter next season's Europa League instead.
A poor run of form in early spring cost them a run at the title but Dawson, who missed the end of season run-in due to injury, believes Harry Redknapp's side competed well over the entire campaign.
"We finished in the top four and you couldn't ask for much more," Dawson told the official Tottenham magazine. "That was our aim at the start of the season and we achieved that.
"True, we were disappointed to be 10 points clear in third place and then to have that steadily eroded and overtaken.
"But we won a lot of tough games and picked ourselves up again after our disappointing spell to really make it start happening for us again, so we can't be too hard on ourselves.
"We played some of the best football in the Premier League at times and that deserves respect."
Former Nottingham Forest defender Dawson suffered an injury-hit season and missed out on the chance to represent England at Euro 2012 but is now hoping to be 100% fit for the new campaign with Spurs.
"The campaign started well for me but then I then had an Achilles injury early on and that needed an operation," he said.
"I was back in action before Christmas and feeling fit and fantastic, but then in March I was unlucky again.
"The injury itself was just a freak. It was just one of those things that happen - there's nothing you can do about it - but I knew as soon as it happened that it was serious.
"I tried to get up again and when I failed I realised I was in trouble. I knew straight away that England would not be an issue for me this summer.
"Now all I want to concentrate on is getting myself fully fit because I'm already looking forward to pre-season."
Meanwhile Spurs winger Gareth Bale has hailed the impact young full-back Kyle Walker has had since breaking into the side.
Walker, who was named PFA Young Player of the Year, has also missed out on England's campaign in Poland and Ukraine through injury but Bale believes he has impressed in his first full season at White Hart Lane.
"He needed a run in the team, he got that and he's showed exactly what he can do this season," Bale told Spurs TV.
"He's played really well. He's set himself a benchmark this season and hopefully he can maintain it.
"Players know all about him now, they will be looking to stop those forward runs and he has to find a way around it - I'm sure he will."