Watson, 21, struggled after being diagnosed with glandular fever earlier this season, going out of both the French Open and Wimbledon in the first round.
But she has since split with coach Mauricio Hadad and showed good determination to beat the qualifier 6-7(5) 6-4 6-3 in two hours and 10 minutes.
Mueller clinched the first set on a tie-break, but Watson managed to break her opponent three times afterwards to set up a second-round clash with French fourth seed Alize Cornet.
Top seed Angelique Kerber beat qualifier Irina Falconi 6-2 6-3, while Mona Barthel saw off Stefani Voegele 6-2 6-2.
Washington Open results
3-Ekaterina Makarova (Russia) beat Eugenie Bouchard (Canada) 6-4 6-2
8-Madison Keys (U.S.) beat Michelle Larcher de Brito (Portugal) 7-5 6-3
Caroline Garcia (France) beat Anna Tatishvili (Georgia) 0-6 7-6(5) 6-4
5-Sorana Cirstea (Romania) beat Lesia Tsurenko (Ukraine) 7-5 1-1 (Tsurenko retired)
Heather Watson (Britain) beat Alexandra Mueller (U.S.) 6-7(5) 6-4 6-3
Monica Niculescu (Romania) beat Taylor Townsend (U.S.) 6-3 6-0
1-Angelique Kerber (Germany) beat Irina Falconi (U.S.) 6-2 6-3
6-Mona Barthel (Germany) beat Stefanie Voegele (Switzerland) 6-2 6-2
4-Alize Cornet (France) beat Yanina Wickmayer (Belgium) 6-1 6-3
- Sports & Recreation
- Heather Watson
- Irina Falconi