However, an 80th minute Gylfi Sigurdsson goal enabled the North Londoners to keep their top-four hopes alive with two matches remaining and thrilled Dawson who says Champions League football is now in their own hands.
"It was a tough battle. We gave ourselves a mountain to climb by giving away a goal so early, but Ade's got us back in it with a great finish," he told Sky Sports 1.
"They went ahead again, but we showed fighting spirit, and kept going and going, then Gylfi's got a great finish. We nearly went on to nick it. They had a few chances, but we showed some spirit.
"It keeps it going. It's out of our hands again if Arsenal win their two games, but we'll keep fighting.
"We've got Stoke at the weekend and hopefully we'll get back to winning ways."