The Mystics win snapped two streaks: Washington's 0-3 record in July and Indiana's five-game win streak.
Meesseman added 13 rebounds and four assists, while Sanders tied a career-high in points and set a career rebound mark with 10 boards.
The Mystics (7-6) took over in the third quarter, as Stefanie Dolson's jumper and follow-up at the end of the period gave them a 22-point lead. The Fever (8-7) managed just six points in the quarter and never recovered.
Tamika Catchings, who entered play averaging 16.6 points, missed all seven of her shots and finished with two points. Marissa Coleman led Indiana with 11 points.
Sanders scored the Mystics' last six points of the first half, including a layup with 0.6 seconds left, to give Washington a 34-29 lead at the break.