A 2-2 draw with Everton on Sunday prevented Spurs from leapfrogging Chelsea into third and the two are now level on points – though Chelsea has played a game less.
Tottenham started brightly, scoring in the opening minute before an Everton rally almost saw the Toffees leave White Hart Lane with all three points until Gylfi Sigurdsson equalized late.
"We showed good character to come back," Dawson told talkSPORT. "We had chances. We hit the bar and at 2-2 we thought we could go on and win it."
He added, "It's always a tough game [against Everton]. David Moyes has a good team there that work their socks off. We'll take the point after being down but we've got to get back to winning ways."
Spurs’ next Premier League game is at home to champion Manchester City, which ran out a 5-1 winner at White Hart Lane last season, and Dawson has called for a much better performance this time round.
"It was our first home game [last season]," Dawson stated. "It was an awful start. We're going to have to be at it [to beat them]."