The Scot ended the match, which was postponed because of heavy rain Tuesday, by executing a drop shot which bounced on the top of the net before falling onto Robredo's side of the court for the winner.
Read: Murray 'happy and lucky' to be father
But there was disappointment for French Open winner Stan Wawrinka who was unable to carry on due to apparent back problems, with the Swiss quitting to hand a 6-7 (8/10), 6-4, 4-0 victory to Australian Nick Kyrgios.
Wawrinka, who did not play after Wimbledon due to a shoulder injury, called for the trainer as he trailed two breaks in the third set.
After speaking with the trainer and his coach Magnus Norman, the 30-year-old third seed decided to give up the match.
Murray set up a meeting with Gilles Muller -- whom he has defeated twice this season -- as he finished his second round contest with 24 winners and 30 unforced errors and breaking four times.
The pair resumed play Wednesday after standing 4-4 in the opening set when the rain stopped play on Tuesday night.
"It was a very tough match, there were a lot of nerves near the end of the first set," said Murray. "I won two close games to take it.
"Conditions today were a lot different, there was a lot of wind. It was difficult to attack and play aggressively. I had to chase down a lot of balls and got lucky a few times as well.
"It was a long day hanging round with the rain, I'm glad I managed to finish it off in two because it's been quite a long couple of days."
Read: Djokovic starts US Open buildup with Montreal win
Rafael Nadal made an encouraging start to the summer hard-court campaign as the Spaniard recovered to defeat Sergiy Stakhovsky 7-6 (7/4), 6-3.
The seventh seed had to wait for two hours before his match at the Uniprix stadium could begin, after a morning rain shower was followed by a towel-drying effort and repairs to bubbles on the rubberized surface.
But once down to business, Nadal made the most of his opening contest on the cement, winning the first set in a tiebreaker and sweeping the second to reach the third round and a date with Mikhail Youzhny, who beat French ninth seed Gilles Simon 6-1, 6-4.
Nadal is playing in Canada for the first time since winning his third title here in 2013. The world number nine improved to 40-12 on the season.
"I go day by day. Today is a victory that I am happy with," said Nadal.
"I'm satisfied with the way that I played. I'm practising well and practising more than ever now because I am healthy and because I feel that's what I need today.
"I won in Hamburg a couple weeks ago, that was an important victory. Six consecutive wins hasn't happened for me very often during the season. That's a positive moment."
Japanese fourth seed Kei Nishikori, the winner of the Washington title last weekend and now ranked fourth on the ATP Tour, started 6-3, 6-3 over Spain's Pablo Andujar in 70 minutes.
Nishikori made a successful comeback last week after a long-running thigh injury and said that his fitness is good.
"I've been feeling pretty good. After winning the title, you have to feel good. My tennis is getting better," he said.
"I didn't start quite well, but it was windy. We had a little wait, so it wasn't easy in the beginning. But after a couple of games I was playing well, concentrating well, too.
"I'm very happy with my tennis today."
Donald Young surprised fifth seed Tomas Berdych 7-6 (7/5), 6-3 while tenth-seeded holder Jo-Wilfried Tsonga had to fight to survive 5-7, 7-6 (8/6), 7-5.
American Jack Sock sprang an upset on troubled 14th seed Grigor Dimitrov, who went down 5-7, 7-6 (7/5), 7-5.
Over recent weeks, the Bulgarian once tipped as a future Roger Federer, lost both his Australian coach Roger Rasheed and Russian girlfriend Maria Sharapova.
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