The NL East-leading Nationals took three of four from St. Louis and passed their 80-win total from last season.
Strasburg, rocked for seven runs last Tuesday at Miami, left with a 2-0 lead and a 2.94 ERA. He threw 97 pitches and wound up with a no-decision.
The 23-year-old Strasburg has struck out a league-leading 195 in 156 1-3 innings this season. The Nationals have repeatedly stated they intend on shutting him down in his first full season since undergoing Tommy John surgery in 2010.
The Nationals have been vague about an innings limit, but the team sat Jordan Zimmermann at 161 1-3 innings last season in a similar situation.
Nationals manager Davey Johnson said on Thursday that Strasburg is probably down to "two or three" starts, meaning the final one could come Sept. 7 at home against Miami or Sept. 12 at the New York Mets.
Washington responded with four consecutive two-out singles against Lance Lynn (13-6). Desmond's flare to center scored pinch-runner Eury Perez with the go-ahead run. Espinosa had three of the Nationals' 13 hits.
Lynn pitched one inning, allowing two runs on four hits for the Cardinals, who have lost five of six while trying to hold off Pittsburgh for the final NL wild-card berth.
Facing the Cardinals for the first time, Strasburg gave up Matt Holliday's two-out double in the first. The All-Star ace held St. Louis without another hit until Holliday's sixth-inning single.
Strasburg walked one overall. He fanned at least two hitters in four innings and regained the NL strikeout lead by whiffing St. Louis pitcher Jake Westbrook in the fifth.
Third baseman Ryan Zimmerman's throwing error, a walk and a wild pitch put runners on the corners with one out in the fourth. Holliday attempted to score the game's first run on Bryan Anderson's shallow fly ball to right, but Bryce Harper easily threw him out at the plate for the final out.
Usually in center, Harper shifted to right with Jayson Werth scratched from the lineup minutes before the first pitch.
Suzuki, acquired in a trade with Oakland last month, stroked his third homer on the season in the bottom half. In the sixth, he tacked on a third consecutive single against Westbrook, which scored Desmond and ended Westbrook's day.
Westbrook allowed two runs on nine hits with six strikeouts and one walk. He avoided his third loss in four starts when Descalso, the Cardinals number eight hitter, turned on a 3-1 pitch from Burnett for his fourth home run of the season.
NOTES: Strasburg also went 1 for 2. He is hitting .277. ... There was no reason provided for Werth's removal from the lineup. He left Saturday's game in the ninth inning with leg cramps. ... The Nationals open a four-game series Monday with Ross Detwiler facing Jeff Samardzija and the visiting Chicago Cubs. ... St. Louis returns home Monday and has Joe Kelly facing Mets starter Collin McHugh.
Link to boxscore: Washington 4, St. Louis 3