Second-half substitute Jarrett notched his second goal of the season to condemn Grimsby – who lost 3-0 at Burton – to non-league football next season.
The Bees started the brighter of the two sides, forcing a number of early saves from Dale goalkeeper Frank Fielding.
Fielding had to be alert to deny John O’Flynn from close range after nine minutes, while on-loan Ipswich midfielder Edward Upson’s 42nd-minute header hit the bar as the home side pressed.
Barnet started the second half in the same fashion as O’Flynn went close again after 55 minutes before Upson saw an effort cleared off the line just four minutes later.
Jarrett then settled the game, smashing home an unstoppable strike from 18 yards.