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The Scotish side paid the penalty for disciplinary indiscretions as the dead-eyed O’Meara punished fired over seven successful shots at goal.
The only consolation for the Borders was a brace of touchdowns from centre Jono Stuart which had the home side eyeing a morale-boosting win at one stage.
It was only after O’Meara has scored three penalties that Stuart reduced the deficit by driving over in the corner on the stroke of half-time following a precise pass from Chris Cusiter.
Hopes of a rare home win were raised on the restart when Stuart conjured-up a carbon his first score as Leinster showed signs of wilting.
But O’Meara kept the visitors in touch with another penalty before the Irishmen regained the lead when Aidan Kearney touched down.
O’Meara slotted another three penalties to open up a 10-point gap before Campbell Feather notched a consolation try deep in injury-time.