With the game into the second minute of added time, a cross into the box from Nikita Parris found Williams, and her header back across the goal beat Villa keeper Daphne van Domselaar to snatch the winning goal.
"It was a top match, we fought to the very end. We are very happy to score the two goals and come back from behind. It was hard work but we just kept going," United's other scorer Lucia Garcia said.
Last season's WSL runners-up United fell behind in the 76th minute to a Rachel Daly goal. Daly, top scorer last season, controlled a pass from Adriana Leon before firing past keeper Mary Earps.
Villa's goal came just two minutes after they had Kirsty Hanson sent off for a high tackle on Hayley Ladd, but their lead lasted just three minutes.
A scramble from a United corner kick saw the ball fall to Garcia and the Spaniard's first-time shot found the net. With the extra player, United knew they still had time for a winner.
As with all games on the opening day in the WSL, the players wore black arm-bands and a one-minute silence was observed for Maddy Cusack, the Sheffield United player who died on September 20th.
