Spurs let a seven-point lead from March slip as Arsenal secured the final Champions League spot on the last day of the Premier League season for the second year running
Spurs went into the game needing to beat Sunderland and for Newcastle to take points off fourth-place Arsenal, despite once holding a seven-point lead over their local rivals in early March.
Arsenal went ahead early in the second half against Newcastle before fans at White Hart Lane appeared to think the Magies had equalised, with the resulting cheers incorrectly informing Dawson & Co. on the pitch, who only found out their true fate after the final whistle.
"We can't just look at today," Dawson told reporters. "We put ourselves in this position. We hoped Arsenal would slip up but they didn't unfortunately."
He added to Sky Sports: "The season we've put in, the work rate, the quality we've shown and to miss out the way we have is devastating.
"At one point I heard our fans cheering so I thought Newcastle had done us a favour and then Gaz [Bale] pulls another world-class goal out. I was on my knees and then I hear it hasn't happened. It's a hard day to take again."
Andre Villas-Boas, meanwhile, has called on his Tottenham squad to use their improvements this term as a platform to push on next season, but admitted disappointment after registering a record points haul in the Premier League and not claiming a top-four finish.
"We wanted to make that final push," the Portuguese said. "Obviously it didn't depend on us and Arsenal did their job and managed to sneak the fourth spot.
"The fact we managed so many points but did not make [the top four] shows the level has gone up. It serves as a reference for next season.
"We have done better than last season in terms of points, but not in terms of finish. We have to improve our level once more [next season]. It is a good lesson, but it's a pity to miss it as [our points return] would normally be enough to qualify."