Southend began the day fifth, but have now leapfrogged their opponents thanks to Worrall's 25-yard free-kick 16 minutes from time.
Phil Brown's side are now behind Wycombe Wanderers on goal difference, while Bury are two points further back.
It was a disappointing result for the home faithful, which numbered more than 8,000 after Bury allowed free admission following the original fixture's abandonment back in March due to heavy rain.
In the Championship, Millwall are staring relegation in the face after a 2-0 defeat at Blackburn Rovers left them four points adrift of safety with two matches remaining.
Bidding to replace Millwall in the second tier are Preston North End and MK Dons, who were both victorious on Tuesday.
Second-placed Preston overcame Notts County 3-1 at Meadow Lane, but MK Dons remain a point behind them following their 3-0 home success against Doncaster Rovers.
County's defeat leaves them two points from safety, level with Leyton Orient, who lost 1-0 at Rochdale.
Swindon Town's hopes of automatic promotion are effectively over after they were held 3-3 at home by Walsall, while elsewhere Scunthorpe United drew 1-1 with Bradford City.
