Trailing by 31 points to zero with half an hour played, the Scots fought back in the second half and took the lead with time running out before George Ford salvaged a draw for the hosts.
Townsend told ITV Sport: “We showed a lot of character today. To respond in the way we did in the second half, I was more than proud.
“We set a goal of winning the second half, not winning the game - we thought that was gone when we were 31 points to seven down.
“But it’s amazing what can happen in a game of rugby. Momentum can swing and we can get confidence and we can take the game to the opposition like we did in the second half.
“And ultimately we’re the team that’s really disappointed to have drawn the game.”