Linehan was 23-of-39 passing for 315 yards including second-half touchdowns of 34 and 23 yards to Deon Watson.
The Vandals (3-4, 2-2 Sun Belt), who beat Troy 19-16 last Saturday, won back-to-back games for the first time since Oct. 2009. Their three wins are the most in a season since 2010.
Louisiana-Monroe (1-6, 0-3) tied it at 13 with 9:03 left in the third quarter when Craig Ford kicked his second field goal, a 31-yarder.
Idaho responded with a nine-play, 79-yard drive capped by Watson's first touchdown less than three minutes later.
The Warhawks went three-and-out on their next two possessions before turning it over on downs on their final two drives.