The 28-year-old wide receiver, who has been with Tampa since being selected as the seventh overall pick in the 2014 NFL Draft, claimed the record in the third quarter when he caught a five-yard pass from Tom Brady.
Evans now has one more touchdown than Mike Alstott, who held the franchise record for almost 20 years.
Chris Godwin and Ronald Jones also scored touchdowns for Tampa Bay while their defence limited the Giants to just 215 yards as the Bucs returned to winning ways after successive defeats.
Victory also lifted the reigning Super Bowl champions (7-3) two games clear of the New Orleans Saints at the top of the NFC South division.
The Giants, whose only touchdown at Raymond James Stadium came from offensive tackle Anthony Thomas, have now lost seven of their 10 games this season and prop up the NFC East.