Captain Immobile, who also had a goal ruled out for offside just after the break, smashed home a 58th-minute penalty to take his top-flight tally with Lazio to 144, one more than Italian goalscoring icon Silvio Piola.
Piola is Serie A's all-time leading scorer with 274 goals over a 25-year career in which his biggest honour was the 1938 World Cup for his national team.
He played at Lazio for nine of those years and scored 149 times overall for the Roman club, another record broken towards the end of last season by Immobile, who is now on 176 Lazio goals in all competitions after netting a league-leading 21st of the season.
"I need to say thanks to the team, in every match and every season they're by my side, helping me to beat each record," Immobile told the press.
"All that you have once you finish playing is how people feel about you. I'm really proud of what I'm doing here, and I thank the fans, who've truly supported me from my first day here."
Since Immobile joined Lazio in the summer of 2016 only Lionel Messi and Robert Lewandowski in Europe's top five leagues – England, Spain, Germany, France and Italy – have scored more goals than the Italian international.
The 32-year-old's winner came on the evening Lazio honoured with a pre-match silence another important captain in Pino Wilson, who led the club to their first league title in 1974 and died aged 76 just over a week ago.
The goal was especially sweet as it allowed Lazio to move above Atalanta and most importantly Roma ahead of Sunday's capital city derby.
Maurizio Sarri's side are two points ahead of their local rivals after one defeat in their last eight matches, which came in the dying seconds against title-chasing Napoli a fortnight ago.
"In the week leading up to the derby I bump into a lot of Roma fans who ask me not to play!" joked Immobile.
Venezia meanwhile stay in the relegation zone, three points from safety after a third straight defeat and the 16th of their first season in Serie A in nearly two decades.