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Top seed Djokovic had to fight back as his game went awry briefly in the opening set on the way to a second-round win over Argentine Juan Monaco 7-5, 6-2.
Second-seed Murray meanwhile kick-started the run-up to his US Open title defence with a resounding 6-2, 6-3 defeat of Mikhail Youzhny.
Murray, the world number two who was eliminated a week ago in the Montreal third round, made sure the mistake was not repeated.
He needed just 70 minutes to see off his Russian opponent and move into a match with Julien Benneteau of France.
Djokovic admitted to a lapse against Monaco, saying: "Between 4-1 and 4-5 were four quite bad games for me."
Djokovic is trying to complete an unprecedented career haul of all nine Masters 1000 titles and only needs to win here in order to fulfil the dream.
"I stopped moving my legs, I didn't really go for my shots," he said.
"I let him dictate the play and wait for his mistake, which was not the right tactics. It was the first match here in Cincinnati, which has different conditions from what we played (last week) in Montreal."
Djokovic eventually sorted himself out and will next square off against Belgian qualifier David Goffin, who defeated Canada's Vasek Pospisil 7-5, 1-6, 7-6 (8/6).
Murray lost his only service game against Youzhny midway through the second set but did not show alarm with the comfort of another break in hand. He claimed victory on a forehand error.
"I put a lot of pressure on myself to get through this one," Murray said. "With the US Open only a couple of weeks away, I need to stay in form."
Murray has twice won the Cincinnati event, both times at the expense of Djokovic, in 2008 and 2011.
Argentine seventh-seed Juan Martin del Potro put out Nikolay Davydenko 7-5, 7-5 while American John Isner set up a duel with another big hitter in Montreal, finalist Milos Raonic.
Isner beat French eighth-seed Richard Gasquet 7-6 (8/6), 6-2 while Raonic defeated Janko Tipsarevic 6-4, 7-6 (7/4).
Women's top seed Serena Williams overcame early jitters similar to those of Djokovic when she lost the first set to Canada's Eugenie Bouchard before winning 4-6, 6-2, 6-2.
The American said she produced the worst shot she can ever remember at one stage in the one hour, 47 minute match.
"I even had an easy approach shot and it went not slightly wide, it was like 50 feet out," Williams said.
"That was actually embarrassing. I thought I'd never hit a shot like that professionally."
Number six Errani defeated Polona Hercog 6-4, 5-7, 6-4 while compatriot and 12th seed Roberta Vinci won a struggle with German Andrea Petkovic 6-4, 6-7 (3/7), 6-3.
Petra Kvitova, the 2011 Wimbledon winner, beat New Zealand's Marina Erakovic 6-4, 6-7 (7/9), 6-3, German ninth seed Angelique Kerber stopped Alisa Kleybanova 7-6 (7/3), 6-2 and former number one Caroline Wozniacki defeated Monica Niculescu 6-4, 6-3.
