Accrington's home win on Tuesday night moved them off the foot of the table and out of the drop zone.
Jeffers, playing for the league's bottom club exactly 10 years to the day since he appeared for Arsenal against Valencia in the Champions League, put Stanley two up with goals in the fifth and 19th minutes.
Will Hatfield added a third before the break and Lee Molyneux completed the rout.
The win took Accrington two points above the bottom two in the desperate fight to stay in the league.
It was a good night for the sides battling to stay up, with Torquay, Plymouth Argyle and Barnet, three of five sides level on points at the bottom at the start of the day, also winning.
Torquay won 2-1 at home to Chesterfield thanks to two goals in the opening eight minutes from Joss Labadie and Rene Howe.
Liam Cooper pulled one back in the 77th minute, but United held on at Plainmoor.
Argyle had to wait a little longer to make the breakthrough at Southend, but a Paul Wotton penalty after 57 minutes and a Reuben Reid finish after 75 earned them a 2-0 victory at Roots Hall.
Barnet were also 2-0 winners, at home to Fleetwood, thanks to a goal in each half from Mark Byrne and Luke Gambin at Underhill.
It all meant Aldershot and York, not in action on Tuesday night, slipped into the relegation places.
At the other end of the table, league leaders Gillingham had to come from behind to draw 1-1 at Rochdale, Danny Kedwell's 56th-minute header cancelling out George Donnelly's opener at Spotland.
Capital One Cup finalists Bradford boosted their hopes of getting into the play-offs with a 1-0 win at home to Wycombe, Garry Thompson the scorer after seven minutes at Valley Parade.