The victory at Windsor Park meant manager Michael O'Neill eclipsed the achievement of Billy Bingham, whose Northern Ireland side twice went nine games undefeated in the 1980s.
Man-of-the-match Washington struck after 41 minutes with his first international goal in his second appearance for the country to become the first Northern Ireland player to score on his home debut since George McCartney in 2001.
The Queens Park Rangers striker outmuscled Miral Samardzic and then barged Nejc Skubic out of the way before cutting inside Bostjan Cesar and firing a low shot past Jan Oblak.
Veteran keeper Roy Carroll, 38, preserved the home side's lead when he saved Milivoje Novakovic's penalty in the 66th minute after Jonny Evans had hauled Samardzic to the ground.
Asked about the unbeaten record, O'Neill said: "It's lovely to achieve something like that. It's for the squad -- to go unbeaten for 10 games is a great achievement."
The 23-year-old scorer Washington, who was at non-league St Ives Town only five years ago, qualifies for Northern Ireland through his grandmother but had never actually visited the country until he was called up for this month's friendlies.
"He's a raw striker. He's come on the international stage and showed what he can do. He made a goal out of nothing and it's a great night for him," added O'Neill.
"It would be unfair to say if he's on the flight to France. He's done himself no harm at all, not only in the performance, but how he's fitted in with the group. It looks like he's going to be an asset to us."
The Irish host Belarus and visit Slovakia in their last two friendlies before starting the Euro 2016 campaign against Poland and then facing Ukraine and world champions Germany.
Slovenia failed to qualify for the tournament in France after losing a qualifying playoff to Ukraine.
(Writing by Ken Ferris; Editing by Mark Trevelyan/Susan Fenton/Rex Gowar)
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