Argyle took the lead through on-loan Alex MacDonald in the 50th minute when he slid the ball under Southend goalkeeper Daniel Bentley after latching on to a pass from Guy Madjo.
Southend's response was almost immediate, with half-time substitute Ryan Leonard sending Assombalonga away down the right. The Southend striker made a brilliant turn - losing his marker with a deft drag back - before racing away to smash the ball between the legs of Argyle keeper Jake Cole in the 55th minute.
Southend substitute Freddy Eastwood fired just over after being set up by goal-scorer Assombalonga as United pressed for a winner.
Argyle manager Carl Fletcher's response was to make a double substitution and introduce fit-again Welsh striker Rhys Griffiths and fellow forward Paris Cowan-Hall.
Cowan-Hall's shot from 10 yards was cleared off the line by Michael Timlin. Southend were reduced to 10 men following Barry Corr's 82nd-minute red card, leaving visiting boss Paul Sturrock having to settle for a point on his return to former club Argyle.