Nationals second baseman Stephen Drew (10) with one of his three hits against the Cardinals on Monday night. Drew also drove in four runs in Washington’s 14-6 win. (Katherine Frey/The Washington Post)

Dusty Baker proposed a simple solution to the Washington Nationals’ early-season bullpen troubles Monday afternoon and it had nothing to do with his maligned group of relievers. The bullpen, the Nationals’ manager explained, can perform better, but it wouldn’t always have to perform in pressure-packed situations if the offense busted out with big innings when given the opportunity.

The season was young, but the Nationals, Baker lamented, were squandering chances to blow out opponents.

“Let’s score more,” Baker implored.

A few hours later, his Nationals obliged, posting their highest run total since August in a 14-6 victory over the St. Louis Cardinals to open a three-game series at Nationals Park. The Nationals didn’t capitalize on every scoring opportunity, but steady pestering from a deep lineup refusing to concede at-bats eventually produced a seven-run eighth inning and the breathing room Baker sought.

The Nationals compiled 19 hits — 11 off Cardinals starter Adam Wainwright, who allowed six runs (five earned) and threw 96 pitches across four innings — and had the leadoff batter reach base in seven innings without their usual leadoff batter in the lineup. They went 9 for 19 with runners in scoring position and didn’t need a home run to put post a couple of touchdowns, extra points included, and overcome four errors in the first five innings.

“It was kind of an ugly game,” said Baker, who passed Jim Leyland for 16th on MLB’s all-time managerial wins list with 1,770. “But when you score, it can cure most of the ills.”

The Nationals (4-3) devoured Cardinals pitching without Trea Turner in the lineup for the second straight game. Turner was placed on the 10-day disabled list Monday, retroactive to Sunday, with a strained right hamstring. He and Baker insisted the ailment isn’t serious.

“I can’t rush back,” said Turner, who had not previously dealt with a hamstring injury. “I can’t do anything stupid. I think it’s good for me because I would definitely try to talk my way back in.”

Without Turner, Stephen Drew slotted in at shortstop and Adam Eaton, who starred as a leadoff man with the Chicago White Sox the past three seasons, moved to the top of the order. The offense didn’t skip a beat.

Eaton recorded three hits and three runs batted in. Drew went 3 for 4 with four RBI, the most he has accumulated in a Nationals uniform. Bryce Harper tied a career-high with four hits, including a bunt single, plus two walks, three RBI, and throwing out a base runner from right field for a double play to end the seventh inning. Ryan Zimmerman compiled three more hits, and Matt Wieters had two. Daniel Murphy registered a single to extend his hitting streak to seven games and cracked three other balls at fielders.

“There’s a lot of talent in this room,” Eaton said, “and being able to see it materialize is great for us.”

The offensive attack masked an unusual bout of defensive sloppiness, which made Tanner Roark’s second start of the season more demanding than necessary. The Nationals’ infield blundered for the cycle; each member, except catcher Wieters and any of the four pitchers, made an error. A couple of other gaffes unnoted in the box score were also costly.

Roark was not himself in the first inning of his first start last week, displaying uncharacteristic wildness after not pitching in 11 days. That wasn’t the case on normal rest Monday. Pumping in fastballs that touched 96 mph — a couple of ticks faster than usual — Roark needed just 14 pitches to dismiss the Cardinals. But the rest of his outing wasn’t as crisp.

With two outs, first base open, and Wainwright on deck in the second inning, the Nationals could’ve elected to pitch around Kolten Wong, if not intentionally walk him. Roark, however, has said he prefers to not pitch around batters, and Wainwright isn’t your typical weak-hitting pitcher. He boasts a .200 career batting average and went 1 for 2 in his first start. So Roark pitched to Wong, and Wong smashed a hanging 2-2 slider off the wall in right field for a two-run double.

St. Louis (2-5) manufactured a third run in the third inning with some help from the first of the Nationals’ defensive mistakes. Dexter Fowler led off with a grounder to Zimmerman’s right, and it went under Zimmerman’s backhand attempt. He then stole second base and scored on Stephen Piscotty’s two-out single.

More defensive miscues — errors from Murphy and Rendon plus another play Murphy could have feasibly made — sabotaged Washington in the next half inning. It could’ve been a disastrous frame, though, if not for Drew’s ensuing diving stop in the hole on a smoked one-hopper from Jhonny Peralta to birth an improbable double play, which required a replay review after the first base umpire ruled Peralta safe at first base.

“Play of the game,” Roark said.

The timely sequence ended Roark’s night. He allowed five runs (three earned) and threw 104 pitches over five innings. Matt Albers replaced him to make his Nationals debut with Washington nursing a 6-5 lead. He stabilized things with two scoreless innings as Washington’s bullpen held St. Louis to one run over the final four frames — a meaningless Jedd Gyorko home run off Shawn Kelley in the ninth inning — while the Nationals scored eight. For once, the other team’s relief corps was getting pounded. It was exactly what Baker ordered.