Whelpdale latched on to a deep free-kick from Joe Martin and, with Shrimpers goalkeeper Paul Smith rooted to his goal-line, he cushioned the ball home from close range.
The Gills came close to doubling their lead when a looping effort from Myles Weston was tipped to safety by Smith, who also did well to keep out a 12-yard shot from former United man Adam Barrett.
However, Southend improved in the second half and almost levelled in the 58th minute when a stunning 30-yard shot from Kevan Hurst was impressively tipped over the crossbar by Stuart Nelson.
Meanwhile, first-half goals from Joe Anyinsah and Matt Lund gave struggling Bristol Rovers a New Year's lift with a 2-1 victory over Plymouth in John Ward's first home game since returning as manager.
The home side went in front after 19 minutes when Argyle goalkeeper Rene Gilmartin could only palm out an inswinging corner from Fabian Broghammer and Anyinsah fired the ball pack past him with a fierce drive from 12 yards.
Another Broghammer corner from the opposite left side saw Lund make it 2-0 with a stunning volley after 25 minutes, while all Plymouth could offer in the first half were a couple of off-target headers from Onismor Bhasera and Nick Chadwick.
The visitors improved after the break and Conor Hourihane shot them back into the game from Durrell Berry's right-wing cross after 62 minutes.