After 25 seconds Sam Hoskins tapped into an empty net when Filipe Morais' shot was blocked and fell kindly to him, with Marcus Haber heading home unmarked in the final minute.
Smith said: "It was a very tough night to play football. When there is a bit of edginess about the group - and there has been at home - to find the back of the net as quickly as that and to start so brightly was not only pleasing but settling for the players.
"There was a determination that probably we haven't seen for a little while here.
"I thought it was a real all-round good performance and capped off with a good goal late on to seal it. It was a real competitive game. They are a good side Notts County.
"We could have made a little bit more of our possession but didn't quite find the right pass. It was always going to be a tough night, our form has been tough and edgy here and we were against a tough side."
Smith also had praise for goalscorer Hoskins and added: "He has been a good addition but he is a young lad still learning his trade. He is a talented boy and he has given us something different."