A breathless encounter was settled by the impressive young striker after Ashley Young, atoning for a penalty miss, cancelled out Mark Noble's first-half opener from the spot.
Habib Beye was sent off with six minutes to go and the 10 men were undone when Hines struck to lift Gianfranco Zola's side out of the bottom three.
Herita Ilunga and Carlton Cole were both forced off injured but the Hammers started well enough with Brad Friedel kept on his toes by a Mark Noble effort before Rob Green had to show his England class at the other end to touch away a deflected Stiliyan Petrov 30-yarder.
Green also saved superbly from Gabriel Agbonlahor and the striker flashed a header over from a Young cross before former Norwich keeper Green snuffed out a John Carew attempt at his near post.
In first-half injury time, the shot-stopper's hard work was repaid when Beye brought down Hines inside the box and Noble found the roof of the net to open the scoring against the run of play.
Villa got a penalty of their own four minutes into the second half when Steve Bennett controversially felt Manuel Da Costa fouled ex-Hammer James Collins. Young's spot-kick was superbly saved by the outstanding Green to ensure justice was done.
Young showed what he is made of by soon making amends, curling a great shot past Green from the left flank to level the scores soon afterwards.
The drama increased as the final whistle approached with Hines going close twice as the home side searched for a morale-boosting winner.
When Beye was sent off for a foul on Jonathan Spector, his second bookable offence, it did not stop Young firing in a goalbound shot that Green again did superbly to palm away.
And, with time almost up, Hines collected a Scott Parker pass to lash in the winner and spark jubilant scenes amongst the home fans.
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