Automatic runner CJ Abrams advanced to third on a flyout by Lane Thomas before scampering home when Meneses lifted a ball to center that allowed Washington (66-84) to salvage the finale of a three-game series.
Brice Turang's sacrifice bunt moved the tying run to third in the last of the 10th, but Robert Garcia (1-2) intentionally walked Willy Adames before getting Rowdy Tellez to ground into a game-ending double play.
Thyago Vieira (0-1) was tagged with the loss despite providing two innings of hitless relief.
Luis Garcia had two hits, including a homer, for the Nationals, who snapped a five-game skid.
Mark Canha and Victor Caratini each went 2-for-4 for Milwaukee (84-65), which finished with just six hits.
Both teams squandered opportunities to push a run across the plate in the 10th, as Washington stranded runners on second and third before the Brewers left the bases loaded in the bottom of the inning.
The Nationals also had a golden opportunity to break a 1-1 deadlock in the eighth.
Drew Milas hit a leadoff double and advanced to third on Jake Alu's sacrifice bunt. Jacob Young then walked, and left-hander Andrew Chafin entered in relief to face the left-handed-hitting Abrams.
That move paid off, as Chafin got Abrams to ground into an inning-ending double play.
Garcia hit a solo home run in the second that provided Washington with a 1-0 lead that held through five innings. But in the bottom of the sixth, Carlos Santana drove in Canha with a sacrifice fly to draw the Brewers even.
Both starters lasted six innings and logged quality starts. Milwaukee's Brandon Woodruff allowed one run on three hits, while the Nationals' Patrick Corbin gave up one run on four hits.
—Field Level Media