Molina delivered two-run singles in a three-run third inning and a five-run fourth.
Meanwhile, Leake (6-6) cruised through eight innings, allowing just five hits and a run with three walks and four strikeouts.
Leake, whose last win prior to this start was May 24 at the Los Angeles Dodgers, attracted 17 outs via grounders. He induced four double play balls and threw just 93 pitches.
In sharp contrast, Washington starter Tanner Roark (6-6) required 85 pitches to negotiate three innings. Roark allowed four hits and three runs, issuing five walks and fanning two.
Reliever Jacob Turner quickly coughed up five runs in the fourth, reducing the game's remainder into a slog to the final out.
The Cardinals (38-41) won their third straight game and are 5-1 in their last six games.
St. Louis broke the scoring seal in the bottom of the third.
Molina poked a two-run single to center that plated Tommy Pham and Stephen Piscotty, who both walked to start the rally. Two outs later, Greg Garcia added an RBI single that scored Jedd Gyorko.
Gyorko ripped a ground-rule double to right that chased Matt Carpenter home. Molina followed with another two-run single through a drawn-in infield for a 6-1 lead. Rookie Paul DeJong capped the inning with a 426-foot two-run shot to left-center, his sixth homer of the year.
NOTES: St. Louis announced Friday that general manager John Mozeliak is being promoted to president of baseball operations. Michael Girsch is the new GM and vice-president. Both signed contracts through 2020. ... Cardinals LHP Zach Duke (left elbow surgery) started a rehab assignment Friday at the team's Gulf Coast League affiliate, pitching an inning and allowing a hit while fanning one. ... Washington officially put SS Trea Turner (broken wrist) on the 10-day DL and recalled INF Adrian Sanchez from Triple-A Syracuse. The Nationals also transferred LHP Sammy Solis (left elbow inflammation), currently on a rehab stint at Syracuse, to the 60-day DL.