It is only the second game that the Magpies have lost in all competitions this season, with their last being to Liverpool in August.
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But they were second best against the Yorkshire outfit, as Windass struck in the 52nd and 65th minute.
Though Newcastle fans will be left fuming over Sheffield Wednesday's first goal with Windass in an offside position as he swept home from close range.
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But there could be no complaints over his second goal as he coolly converted past Martin Dubravka after being picked out by Michael Smith.
The Toon immediately hit back through Bruno Guimaraes after the Brazilian reacted quickest to Cameron Dawson's save to deny Chris Wood.
Again there was a suspicion of offside, but with VAR not in use at the League One ground, the goal stood.
Newcastle pressed for a late equaliser, but it was to no avail as their wait for a first FA Cup triumph since 1955 continues.
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West Ham also sprung a shock away at Brentford - just a week after the Bees beat David Moyes' men at the London Stadium in the Premier League.
And it was former Brentford star Said Benrahma that struck the hammer blow after a stunning effort from 20 yards out.
Brentford had no answer to his strike in the 79th minute with West Ham showing defensive resilience when needed.
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In another FA Cup shock, Nottingham Forest were knocked out by Blackpool, and National League side Wrexham won 4-3 away at Coventry.