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The Robins should have scored when Nick Colgan could not hold onto a shot, but Tykes defender Matt Carboon hoofed the ball upfield to Hayes.
The former Iron forward turned, and his shot from the touchline flew into an empty net.
Ritchie, who took over at Oakwell towards the end of last season, said: "Paul has got everything.
"He's got strength, because the lad was pulling him back, a bit of pace, his control is good his finishing is great and he is an excellent striker."
Hayes joined Barnsley in the summer on a Bosman transfer, though United are entitled to a fee - which will be set by a tribunal.
And he will make an early return to North Lincolnshire on Tuesday, when United host the Tykes in a League One clash.
Midfielder Richard Kell also made the move from Glanford Park to Oakwell, despite suffering a broken leg while playing for United in the penultimate game of last season.