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Eddie Kinsella's side will now face Kildare in the next round.
It took penalties to decide last night's Eirgrid Leinster Under-20 football quarter final.
It finished Laois 13 points Westmeath 1-10 after extra-time in Portlaoise.
But the home side prevailed 4-3 on spot kicks at O’Moore Park to set up a semi-final date with Kildare next week.
Despite the win, Laois manager Eddie Kinsella was disappointed the game had to go to penalties:
Westmeath boss Damian Gavin says they missed too many chances: