Cruz, who hit a two-run double in the fifth, smacked a 1-1 pitch from Brandon Hughes (1-1) to center to snap a 4-all tie with one out in the eighth. It was Cruz's ninth homer of the season and first since June 25.
"He's been working really hard on getting the ball in the air," Nationals manager Dave Martinez said. "He's hitting the ball but it hasn't really been in the air, and to get that ball in the air and hit it the way he did was awesome."
Cruz is 9 for 25 (.360) over his past eight games.
"It's about adjustments. You've got to adjust on the fly. Even (between) at-bats," he said. "I've been more consistent hitting the ball and swinging at strikes. That's the key."
Ian Happ hit two solo homers for the Cubs, including a drive into the second deck in right center, giving him three in his past three games.
Washington recalled shortstop prospect CJ Abrams from Triple-A Rochester on Monday and he went 0 for 4 with a throwing error in his Nationals debut. Abrams, 21, was one of five prospects obtained from the San Diego Padres in the Juan Soto-Josh Bell trade on Aug. 2.
Starter Josiah Gray allowed three runs on five hits for Washington. He walked two and struck out 10.
Cubs starter Marcus Stroman allowed four unearned runs on six hits in 4 2/3 innings.
Trailing 3-0, the Nationals rallied during a four-run fifth inning started when Lane Thomas reached on Zach Ministry's throwing error. Cruz greeted reliever Mark Leiter, Jr. with a two-run double down the left-field line to make it 4-3.
"I thought (Stroman) threw the ball well," manager David Ross said. "Didn't play good defense that last inning and I probably took him out one batter too soon."
IN GOOD COMPANY
Happ's first homer was his 11th of the season. He joined Mark Teixeira, Jose Cruz, Ruben Sierra, Eddie Murray and Mickey Mantle as the only switch hitters in MLB history to hit 11 or more homers in each of their first six seasons.
"Any time you can have little things like that where you're on a list with some of the best to ever do it, those are really cool moments that I try to cherish," Happ said.
Washington's Joey Meneses went 1 for 4 and has hit safely in 10 of 11 games since his major league debut on Aug. 2. He is 15 for 39 (.385) with five homers, seven RBI and eight runs scored in that stretch.
Cubs: RHP Kyle Hendricks (strained right shoulder), on the injured list since July 6, has been cleared to begin a rehabilitation program in Arizona. ... LHP Wade Miley (left shoulder strain) will make his fourth minor league rehab start on Tuesday.
Nationals: SS Luis Garcia (left groin strain) was placed on the 10-day injured list retroactive to Aug. 13.
Cubs: LHP Justin Steele (4-7, 3.63 ERA) has recorded 19 strikeouts over 10 2/3 innings of his last two starts, both no-decisions.
Nationals: LHP Patrick Corbin (4-16, 7.02 ERA), whose last turn was skipped after he failed to make it through the first inning in two of three starts, is 5-2 with a 3.97 ERA in eight career starts against the Cubs.