Bomar, in his second year out of Sam Houston State, has yet to take a snap in a regular-season game. He became the backup to Eli Manning when Jim Sorgi was placed on injured reserve this week.
The Bomar scoring pass came after the Patriots had taken a 17-12 lead with 4:55 to play on Brian Hoyer's 66-yard throw to Darnell Jenkins.
Manning started the pre-season contest and went 7-of-9 for 91 yards and a touchdown. New England's Tom Brady was 4-of-8 for 51 yards, a touchdown and an interception.
Both teams completed the pre-season with 2-2 records. The Giants host Carolina to begin the regular campaign and New England will be at home against Cincinnati.