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United knew a win would it them level with second-placed Manchester City in the Premier League table, but Wilfried Gnonto stunned the Red Devils by firing Leeds ahead inside one minute.
Raphael Varane's own goal then gave United a mountain to climb, but in-form forward Rashford gave the Red Devils hope with a fine header, setting up a grandstand finish.
Sancho – who recently returned to the fold after a period of absence – levelled things up as a second-half substitute, before United failed to force a winner in a captivating finish.
Leeds came flying out of the traps and scored after 55 seconds as Gnonto drilled into the bottom-left corner following a give-and-go with Patrick Bamford.
The host side then laid siege to Leeds' goal for the rest of the first half, with Alejandro Garnacho volleying wide before Maximilian Wober denied him with a goal-line clearance.
Marcel Sabitzer drew a spectacular save from Illan Meslier from 25 yards, and the Red Devils fell further behind shortly after half-time.
The outstanding Gnonto found substitute Crysencio Summerville on the underlap, and his cut-back was diverted past David de Gea by Varane.
United found a route back into the contest just after the hour as Rashford rose to head Diogo Dalot's cross home, and their pressure told once more with 20 minutes remaining.
Shaw's cut-back deflected into the path of Sancho, who saw his tame effort squirm through Meslier's grip to level things up.
Varane almost went from villain to hero with a powerful header as United continued to press, but Meslier atoned for his own earlier error with a save to preserve Leeds' point.