The Red Devils fell behind in the 11th minute at the Vitality Stadium when Callum Wilson netted for the home side.
However, Manchester United fought back in the 35th minute when Anthony Martial netted for the visitors to draw level.
Rashford was introduced with Ander Herrera in the 56th minute, and the England international netted the winner from close range in the 92nd minute.
And Manchester United legend Scholes felt that the Red Devils were just about deserving of the three points.
Speaking to BT Sport immediately after the game, Scholes said: “I thought in the second half they [Man United] were a lot better.
“Rashford coming on made a big difference and Herrera as well allowed Paul Pogba to be released.
“But give credit to Bournemouth as well, they kept going and created chances.
“Winning the game, was that a fair result? I think in the second half they just shaded it.”
Manchester United’s win lifted them up in to seventh place in the table and left them six points behind leaders Manchester City.
The Red Devils will return to Champions League action when they take on Juventus in Turin on Wednesday night.
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