Spurs' 12 games without defeat towards the end of 2012 meant Andre Villas-Boas' side were odds-on to qualify for the Champions League, but losses to Liverpool and Fulham have allowed their rivals back into the race.
Another defeat at Swansea on Saturday, coupled with an Arsenal victory over Reading, will leave the Gunners just one point behind their north London rivals with a game in hand.
"Everyone was disappointed with the result against Fulham but that's behind us now and we have to push on again," Dawson told Spurs TV Online.
"We showed exactly what we're capable of when we went on that unbeaten run and we've got to look to do something similar between now and the end of the season. We know it will be tough, but the determination is there."
Capital One Cup winners Swansea have little to play for during the remainder of the season but Dawson is refusing to take the Welsh side lightly, especially on home turf.
"Swansea have impressed everyone again this season," Dawson added.
"They play good football and the Liberty is always a tough place to play, but we have to go there and bring back a positive result."
Dawson's caution is shared by fellow defender Steven Caulker, who played 26 times for Swansea during a loan spell last season.
"I had a great year there and I'm excited about returning on Saturday," the England centre-back said.
"They are always a tough team to beat and they have real belief at home but we'll be confident and go there to play our football.
"The gaffer will have his tactics spot on and it will be up to us to deliver."