The O's striker had already issued a warning to the visitors with three first-half efforts that brought saves from Chris Weale, but the Shrews keeper was unable to keep out two clinical Mooney finishes after the break.
The Irishman's first of the afternoon came after 56 minutes following a superb cross by Scott Cuthbert that Mooney headed inside the near post from 12 yards.
Full-back Cuthbert also played a key role in the second goal.
He delivered a long throw into the box that Kevin Lisbie back-headed for Mooney to meet and score from close range, taking his personal tally for the season to three. Then a solo run by Odubajo in the first minute of added time ended with the talented winger sublimely placing his shot into the far corner of the goal.
Earlier, Shrewsbury had resisted by dragging a host of bodies back behind the ball, creating few chances.
The best of those fell to Aaron Wildig, whose fierce shot was spilled by O's keeper Jamie Jones, who recovered quickly to smother the ball.