The run by Hampton was one of the few exciting moments in an otherwise unspectacular game between two teams missing a total of nine starters because of holdouts and injuries.
Quarterback Jeff Hostetler scored on a five-yard keeper for New York, and Giant rookie Matt Stover added field goals of 44 and 49 yards.
The Bills, playing without holdout running back Thurman Thomas and using Jim Kelly for only the first quarter, managed only a 39-yard field goal by Scott Norwood and a 28-yard field goal by Kerry Brady.
Hampton's scoring run came on his first carry of the game in the second quarter after a 10-yard holding penalty gave New York the ball on the 11.
The rookie, who finished with 117 yards in seven carries, took the handoff on a delay and then hesitated at the line of scrimmage to set up his blocking.
A block by Bob Kratch sealed off the right side of the Bills' defense, and Hampton broke free to put the giants up 7-3.
The Giants made it 10-3 early in the third quarter after a drive stalled and Stover kicked a 44-yard field goal.
Hostetler made it 17-3 on a scramble in the third quarter.
The score followed Howard Feggins' interception return that gave the Giants the ball at the Buffalo 22.
Stover's second field goal put the Giants ahead 20-3 late in the third quarter.
Brady's field goal, which closed out the scoring with just under six minutes left, came after the Bills lost seven yards on three plays with the ball inside the Giants' 10.