Tries from replacement scrum-half Fabio Semenzato and No.8 Robert Barbieri got the Biancoverde right back in it before Alberto Di Bernardo kicked a last-minute penalty from 50 yards to seal victory.
The result made it two wins from five for Treviso this season as they put last week's 34-18 defeat at Cardiff Arms Park behind them, and Smith maintains he was always confident his men would be able to dig themselves out of trouble.
"I always had trust. We never believed we were out of the game even at half-time," said Smith.
"At half-time we weren't in a crisis. We made loads of errors in the first half but we played really well in that crucial part of the game towards the end.
"We knew we had the wind behind us in the second half and we changed the manner of our game with the rain that came down.
"Perhaps we started to kick a bit more out of hand because I saw that they didn't want to create much and we had nothing to lose by getting down in their half more.
"I fully believe in the work we are doing now. We are a strong team who last week had a bad game and this week I prepared the team to play rugby for two halves, not just one."
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