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Both sides had opportunities to score in an end-to-end first period, with Van Persie's strong strike kept at bay just before the break.
On 32 minutes United suffered an injury blow, with Nani forced off after grimacing in pain with a groin problem.
The Portuguese winger was replaced by Henriquez, who started brightly in a wide left role.
Things got worse for United just after the break as AIK took the lead through Robin Quaison, who thumped a powerful effort home after being give room to shoot on the edge of the area.
Vidic then went close to turning into his own net, but United managed to respond 20 minutes from time thanks to a strong run from Wilfried Zaha.
His low cross into the box found Henriquez unmarked, with the Chilean making no mistake to tap home from close range.
Danny Welbeck and Van Persie spurned a late chances to hand United victory, with Moyes forced to settle for a draw by the Swedish outfit.