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.
The visitors had a huge let off a minute later when Barry Corr volleyed wide of an open goal from inside the six-yard box.
But, having weathered the storm, Gillingham almost added a second when Weston fired against the foot of the right post from 12 yards.
Derek McInnes wants his Aberdeen players to dare to dream as they look to grab their first win in 10 attempts against Celtic.
Brendan Rodgers is looking for Celtic to move on from their Europa League exit when they return to Ladbrokes Premiership action at Aberdeen on Sunday.