Cueto (2-0) allowed one earned run, scattering five hits. He struck out seven and lowered his ERA to 1.78 for the season.
The game time temperature was 45 degrees.
"He can make the ball sink, he can get it to 94 miles an hour," said Cubs manager Dale Sveum. "In these conditions against those pitchers, it's very tough to string a lot of hits together."
Two runners were on base with one out when Cueto exited in the seventh. With two outs and the bases loaded, Aroldis Chapman struck out Ian Stewart looking with a fastball that registered 99 miles per hour on the stadium scoreboard, preserving the lead.
Chapman also worked a scoreless eighth, though he did walk his first two batters this season. In 10 1-3 scoreless innings, Chapman has 18 strikeouts while allowing just three hits. Former Cub Sean Marshall pitched the ninth, picking up his third save in three opportunities.
The Reds capitalized on Chicago mistakes to break a 2-2 tie in the sixth. Cubs starter Randy Wells and reliever Scott Maine combined to walk two batters and hit one more, while Geovany Soto made two throwing errors on sacrifice bunt attempts.
"Good bunts, but those plays have to be made," Soto said.
All told, the Reds sent eight batters to the plate in the sixth, scoring two unearned runs without the benefit of a hit.
"It's what you call an ugly win, I guess," said Reds manager Dusty Baker. "It was sort of which team played the worst today. We were fortunate enough to come away with the win."
Wells was making his first start of the season in place of Ryan Dempster, who was put on the 15-day disabled list Saturday with a right quad strain. Wells struggled with his command, walking five batters in five innings and throwing just 52 of his 93 pitches for strikes.
"To Stubbs there, I missed with a couple pitches and got to the 3-2 count," said Wells. "I thought I made a pretty good pitch to Votto, but the wind wreaks some havoc.
"I wasn't quite as sharp as I'd like to be, but being the first time out it was exciting."
Wells kept the damage to a minimum, limiting the Reds to two runs and leaving with a no-decision. Rodrigo Lopez (0-1) took the loss in relief.
Chicago stranded 12 runners, while the Reds stranded 13. The teams combined to leave the bases full five times.
"We battled and got on base, we just couldn't get that big hit," said Sveum. "They have (Aroldis Chapman and Sean Marshall) in the bullpen and we tried to mix and match and get some matchups.
"But that's why, like I said going into this series, those are two of the best in all of baseball and it's not that easy to get hits off them."
The Cubs tied the game at two in the fifth inning when Joe Mather scored from third after a missed catch error by Votto on a pickoff throw at first base followed by an RBI single by Brian LaHair.
Starlin Castro singled, tripled and scored two runs for the Cubs, extending his hitting streak to 10 games, the sixth streak at least that long in his young career. He has reached base in 55 of his last 56 games.
Jeff Baker added an RBI groundout in the ninth inning for the Cubs.
Votto doubled twice, walked twice, scored a run and drove in another to pace the Reds' offense. The slugging first baseman went his 14th straight game without a home run.
NOTES: Wells was recalled before the game from Triple-A Iowa, taking the roster spot of Marlon Byrd. The Cubs' starting center fielder was traded to Boston on Saturday for reliever Michael Bowden and a player to be named later. ... Sveum said he plans to "mix and match" to fill the void in center field created by Byrd's departure, using a combination Campana, Reed Johnson and Joe Mather at the position. Campana started on Sunday. ... Reds second baseman Brandon Phillips missed the game because of the lingering left hamstring injury that has caused him to miss six of Cincinnati's last 12 games.