A late Danny Grewcock try was too little too late for Bath, as Irish had already done enough through touchdowns from Sailosi Tagicakibau and Topsy Ojo to record a win that took them into eighth place.
Smith said: "This time last year against Bath we were brilliant in the first half, scored a lot of good tries and went on a run of wins from there, so let's hope this is the catalyst for something similar."
Bath head coach Steve Meehan admitted his side paid the price for their lacklustre first-half display - which meant he was forced to keep Grewcock on the pitch for the full 80 minutes.
Meehan said: "In a perfect world we would have been able to replace him, but the situation meant we felt he needed to stay on the field."
Relegation favourites Worcester cut the gap on nearest rivals Northampton to just three points - despite losing 13-6 against Sixways visitors Leicester and gaining a verbal blast from Warriors rugby director John Brain.
Northampton's 28-15 home loss to Harlequins meant Worcester moved within striking distance, although Saints have a game in hand and Warriors' remaining seven-game league programme includes away appointments with title play-off contenders Bristol, Gloucester and Sale Sharks.
Leicester rested England forwards Martin Corry, George Chuter and Louis Deacon, but Worcester could not take advantage as they suffered a seventh home defeat of the Premiership campaign.
An early Seru Rabeni try proved enough to steer Leicester home, but Tigers head coach Pat Howard admitted: "A lot of our guys hadn't played since the Heineken Cup game against Munster last month and I thought that showed, but we should still be better than that."
Northampton's fourth successive league defeat - they last won in the Premiership three days before Christmas - saw them slip deeper into relegation trouble.
Saints head coach Paul Grayson said: "There is a lot of pressure out there, and that makes people tighten up and make mistakes.
"The mood in training during the week is good, but there is a weight on the players' shoulders. It is easy to talk about playing your way out of trouble, but doing it is another thing."
Paul Tupai and Ben Cohen scored tries for Saints, but Quins eased into the Premiership's top six as touchdowns by Andy Gomarsall, the rapidly-emerging David Strettle and Steve So'oialo ensured a comfortable success.