The game was tied 1-1 in the sixth when Prince Fielder, Victor Martinez and Jhonny Peralta got aboard with two outs and Avila sent a 2-1 pitch from Stephen Strasberg deep into the right field stands for his first career grand slam.
Sanchez (9-7) tamed the Nationals on one run, yielding five hits and a pair of walks. The right-hander tossed 57-of-95 pitches for strikes as the American League Central-leading Tigers won a fourth straight.
Strasberg (5-9) surrendered five runs on six hits over seven full frames.
Ryan Zimmerman drove in the only run for Washington, which dropped the opener of a two-game interleague set.