The vital goal came after 58 minutes when Sean Clohessy sent over a cross and Leonard reacted quickly to guide the ball home, via a deflection off O's defender Mathieu Baudry .
The League Two side almost doubled their lead four minutes later when a 25-yard shot from Kevan Hurst shaved the outside of an upright.
Southend keeper Paul Smith was the busier of the two keepers in the first half, pulling off three superb saves - two from Shaun Batt and one from David Mooney.
But the only real moments of concern for the visitors in the second period came in the eight minutes added on following a serious injury to Southend's Michael Timlin.
Orient piled forward but Smith proved unbeatable and twice denied Mooney, although he was fortunate when the O's striker crashed a shot against the crossbar and then sent the rebound over the top.