first time, yet the Scot was not for turning and wrapped up the first set before 40 minutes lit up on the scoreboard.
Fifth seed Milos Raonic breezed past wildcard Robby Ginepri 6 2 6 2 but two other seeds went home, with number nine Ernests Gulbis beaten 6 4 6 4 by wildcard Steve Johnson and 13th seed Roberto Bautista Agut losing 6 4 6 1 to Gael Monfils.
Murray won the three opening games in less than 10 minutes but Sousa the world number 37 finally held after the advantage changed hands numerous times during the fourth game.
The world number nine, whose Wimbledon title defence ended at the quarter final stage, encountered a major obstacle on his road to redemption when Jo Wilfried Tsonga dumped him out of the ATP Rogers Cup last week.
Murray advances in Cincinnati Express Star
Although Sousa frustrated Murray at 2 1 in the second set and eked out two more Puma Roma Grey
games, the former Wimbledon champion was in cruise control despite the Cincinnati heat and breezed to victory.
The German took the first set but the sixth seed is made of stern stuff and battled back to win 6 7 (4/7) 7 6 (7/4) 7 6 (7/4).
The Swiss, who had a first round bye, needed a tie break to take the first set but then dropped the second against his Canadian opponent before prevailing 7 6 (7/4) 5 7 6 2 to move into the last 16.
There was an exchange of games before Sousa broke Murray for the Puma Mcqueen Trainers
Murray's Wimbledon conqueror Grigor Dimitrov suffered a surprise exit as he lost 6 4 3 6 6 3 to Jerzy Janowicz but 16th seed Tommy Robredo had no such problems as he beat Sam Querrey 6 2 6 4.
and, despite playing in 63 per cent humidity at the Lindner Family Tennis Center, made light work of Portugal's Sousa, winning 6 3 6 3 in little over an hour.
Roger Federer's quest for a sixth Cincinnati Open title at the event got off to an unconvincing start as he needed three sets to get past Vasek Pospisil in Cincinnati.
David Ferrer set up a last 16 clash with Mikhail Youzhny but he needed three tie breaks to win an epic tussle against Philipp Kohlschreiber.
Youzhny had a far easier time of it as he beat Andreas Seppi 6 3 6 4, while Briton James Ward lost 6 2 6 2 to Julien Benneteau.
But the Scot took that setback in his stride Puma Shoes For Men Navy Blue
Tomas Berdych suffered an early exit as the fourth seed was upset by world number 43 Lu Yen hsun. The Czech took the first set and appeared to be Puma All Blue on his way to victory but Lu struck back and took the match 3 6 6 3 6 4.
He will next play big serving 11th seed John Isner, who beat Marinko Matosevic 6 3 7 6 (7/1).
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