Posh will take on Chesterfield in the JPT final at the end of March, but Washington believes that his team could return a couple of months later in the League One play-off final.
"Obviously, the place is buzzing with the lads getting to Wembley," Washington told ITV News Anglia.
"Everyone's really excited and it's a great opportunity for all the lads. Now we've got to fight hard for the rest of the season to hopefully get there again."
Peterborough travel to local rivals Stevenage on Saturday in a game which carries huge significance at both ends of the table.
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Mainly light rain arriving later in the night. On Saturday, mainly cloudy with periods of rain and some bright intervals.
Norwich-based insurance company Aviva has seen a 70% increase in calls and claims for damage since Storm Doris hit.
Storm Doris brought havoc and destruction across the Anglia region almost 30 years on from the Great Storm of 1987.