Spurs went down to 10 men after just 28 minutes when Jermain Defoe received a straight red card after an aerial challenge with James Collins.
"We had to play with two banks of four with Rafa up front for so long and that just shows the spirit we've got," Dawson, outstanding in central defence alongside the returning Younes Kaboul, told the club's website.
"The manager said at half-time that we had to keep going and doing the same as we had been doing.
"We kept the ball well, especially five minutes before the end of the first half, but we didn't keep it as well in the second half.
"We showed the spirit and that was the main thing.
"The squad we have is fantastic. We are going places. When you go down to 10 men at Villa and get three points that is a good sign."