They face Waterford in Cusack Park next.
Wesmeath's position in Division 1B in the National Hurling League is hanging by a thread, after a fourth defeat in four games against midland neighbours Offaly at O'Connor Park over the weekend.
Despite having a strong wind advantage in the first half, the Lake County were unable to make a sizeable dent on the scoreline, leading by just two at the interval.
Offaly ran out 1-22 to 19 point winners.
The result leaves the visitors bottom of the table and now must win their two remaining games and hope other results work out in their favour to stand any chance of staying up.
Westmath manager Seaoirse Bulfin says his side have lots to work on.
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