Luis Cruz hit a two-run single to help the Dodgers pull within three games of idle St. Louis for the final NL playoff spot. Both teams have six games remaining.
"There's not much else to say about it other than we just have to win and hope we get some help," Kemp said.
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The Dodgers open a six-game homestand Friday night with Cy Young Award winner Clayton Kershaw on the mound against last-place Colorado. The Cardinals start a three-game series at home against NL East-leading Washington.
"We have to win every day," Los Angeles manager Don Mattingly said. "Our only chance is to put pressure on St. Louis and make them win every day."
Capuano (12-11) carried a 6-0 lead into the sixth. The left-hander allowed one run and five hits in 5 1/3 innings for his first win since Aug. 12 at Miami.
Cruz had three hits, including his key single in the fifth as Los Angeles scored four runs with two outs to go ahead 5-0.
The Dodgers had 14 hits for the second straight game. Before that, the struggling offense managed more than three runs in only four of 19 games.
"We've been better," Mattingly said. "We've got some big hits and not just from the big guys. This is really more what you kind of expect."
Kemp was 2 for 5 with an RBI to finish the season hitting .406 (26 for 64) with seven homers and 20 RBIs in 15 games against the Padres.
Los Angeles is 29-4 when Kemp has a multihit game this year.
"We're still breathing," catcher A.J. Ellis said. "All we can do is go out and basically try to run the table. That's what we are trying to do. We go [Friday] night and have Kershaw on the mound and I like our chances."
Rookie starter Casey Kelly (2-3) gave up five runs and six hits in 4 2/3 innings. The right-hander struck out six, walked two and hit two batters with pitches.
The loss assured San Diego (74-82) of its fourth losing record in the last five seasons. The Padres have had 13 winning seasons in their 44-year history.
"Obviously, everything is a learning experience," Kelly said. "You take some bumps in the road. I thought I threw the ball better, but had the same problems with command in that one inning."
Ellis' RBI single in the fourth put the Dodgers ahead before they struck for four runs in the fifth.
"It's definitely frustrating when I didn't make the pitches in those two at-bats," Kelly said. "Ahead in the count and hitting those guys, it made it a different ballgame."
Nick Vincent replaced Kelly and gave up Cruz's two-run single followed by Ellis' RBI single. Vincent struck out Capuano to end the inning.
Capuano walked Chase Headley to open the sixth and allowed Yasmani Grandal's single before throwing a wild pitch. After getting an out, Capuano was lifted for Paco Rodriguez, who got the second out before giving up Everth Cabrera's infield single to load the bases.
The Dodgers finished 41-40 on the road, their third winning record away from Dodger Stadium in the last 10 seasons.
- Capuano pitched past the fifth inning for the fourth time in his last eight starts.
- The Dodgers finished 11-7 this season against San Diego.
- Padres 2B Logan Forsythe was scratched from the lineup with a sore right knee.
- Yonder Alonso had three hits for the Padres, including a pair of doubles to run his season total to 37. It was his 10th game with at least two doubles, the most for any major leaguer since 2006 when Pittsburgh's Freddy Sanchez also had 10.
- Giants RHP Ryan Vogelsong (13-9, 5.38) opposes Padres LHP Andrew Werner (2-2, 4.64) on Friday night at Petco Park in the opener of a three-game set.