Paul McConway's side were two point winners.
Tullamore Manager Paul McConway says the ending to the game against The Down's is a learning curve for his side.
The Offaly champions secured their place in the semi-finals of the Leinster Football Championship with a 3-13 to 4-8 in Mullingar on Sunday.
Cillian Bourke's 52nd minute goal left Tullamore 3-12 to 2-4 ahead, with just eight minutes remaining, before the Westmeath side doubled their tally in the final 10 minutes to almost pull off a dramatic comeback.
McConway says his team need to perform for the full 60 minutes:
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