The Mystics (13-15), who have won three straight, led Chicago and New York by 1 1/2 games for the final playoff spot in the Eastern Conference, though both teams were to play later Tuesday night.
Sophia Young scored 26 points for the Silver Stars.
Monique Currie also scored 10 for Washington and Sanford added nine rebounds.
The Mystics lost Nikki Teasley to a sprained left knee 7:33 into the game. Teasley, the team's assists leader and one of five Mystics averaging in double-digit scoring, was carried from the floor and rode a wheelchair to the dressing room after injuring the knee while guarding Shanna Crossley on a fast break.
San Antonio (17-11) clinched a playoff spot in the West last week but fell a half game behind Phoenix for the conference lead.
The teams swapped the lead 13 times and were tied 10 more before Washington went ahead 42-40 on a layup by Sanford 2:50 into the second half.
The Mystics took a 64-53 lead when Sanford followed Beard's missed layup with seven minutes remaining. But the Silver Stars cut it to 70-66 on two free throws from Marie Ferdinand-Harris with 1:32 to go.
Washington was ahead 75-70 with 18 seconds left after Beard's breakaway layup, but Darling hit a 3-pointer with six seconds to cut it to 75-73. The Silver Stars fouled Beard, who hit both free throws to seal the win.
San Antonio missed its first seven shots from the floor to start the fourth quarter. They didn't score until Young made a free throw 3:52 into the period.
With the playoff spot clinched, the Silver Stars played their second consecutive game without Hammon and Johnson. Hammon, third in the league in scoring and first in assists, has a left foot sprain. Johnson missed her eighth game in 340 career contests because of a left Achilles' tendon sprain.