Jim Casey, who researches things like this, tells us that the Expansion Senators' "win of the day" was in 1961:
"Happy to be early with today's Expansion Senators win of the day. From 1961, in the second game of a doubleheader, the Nats top the Yankees 2-1,l behind the brilliant pitching of Hal Woodeshick."
Note:
Attendance: 21,904
Venue: Griffith Stadium
Game Duration: 2:14
22,000 was a huge crowd at Griffith Stadium, which sat about 25,000 for baseball. Some of those were behind poles and pillars, and maybe 7500 or 10,000 were in the bleachers, and the bleachers were somewhere in Prince George's County, it used to seem.
Look at that 1961 Yankee lineup, and drool. The Nats lineup, in the first season of the expansion team ("the New Senators", Washingtonians called them), featured Coot Veal (himself!) at SS. Gene Woodling, who was in RF, had been one of the Yankee "Gold-dust Twins" alongside Hank Bauer. I remember watching Woodling line a ball over the RF wall, a feat since the wall angled from 320 on the RF line out to 425 or more in right-center, and stood about 30 feet high. Dutch Dotterer started at catcher for the Nats, and I vaguely remember the name, but don't remember a thing about him.
https://www.baseball-reference.com/boxes/WS2/WS2196104302.shtml?fbclid=IwAR3zBNugr-BGguu7peIfJT54LBiwDExdqFolDFr65937h5VJV-RrhO2gqRM