The hosts had numerous chances to get on the scoresheet, but an inspired performance from Newcastle goalkeeper Tim Krul saw the visitors come away with the points.
"We were fantastic in the second half but we couldn't get the ball in the back of the net," Dawson told the club's official website. "Tim Krul made some incredible saves. It wasn't our day.
"Everyone was behind us, the atmosphere was fantastic and I'm gutted we couldn't get that first goal to give us a chance to perhaps go on and win it.
"It's a huge disappointment not to have taken anything from that game."
Tottenham return to Premier League action on November 23 when they travel to Manchester City.