Jamal Campbell-Ryce put the home side ahead from the spot after regular penalty taker Jeff Hughes had earlier fluffed his lines.
However, Stevenage deservedly scored a penalty themselves when Lucas Akins finished after Alan Sheehan had handled and subsequently received his marching orders. And it was down to substitute Robin Shroot to make sure Stevenage took away all three points with a dramatic injury-time winner at Meadow Lane.
Having seen his side move up to second in the table, Smith said: "This is a tough place to come, so to get away from here with all three points is a great achievement.
"The boys have had to battle hard for 90-plus minutes and have absolutely worked their socks off and I think for that we deserved to nick one late on.
"There is a great spirit and togetherness in the camp and that's there for all to see with a performance like that. It's a tough league so to be where we are right now is terrific, but there's a long way to go and we know that. But to get a win like this breeds confidence and that's absolutely vital in this game."