Bedene, whose last match was a defeat in the third round of Wimbledon to Gilles Muller, broke early in the first set to move 2-0 ahead, only for both players to then fail to hold as the Briton led 3-1.
The world number 48 soon closed things out in 42 minutes when taking his third set-point chance.
Sousa, ranked 63 in the world, was broken again at the start of the second set, but then saved four break points in game three to trail 1-2 before then breaking back against Bedene to level.
Although Slovenia-born Bedene, who reached the final of the Hungarian Open in April, rallied again to break Sousa and lead 3-2, he failed to hold again in game eight as the Portuguese squared things at 4-4.
Bedene then had to serve to avoid a third set, but the momentum was with Sousa who broke at 15-40 to force a decider.
The 28-year-old regrouped for the final set and forced three-break chances in game three, which Sousa fended off.
The deciding set continued on serve until Bedene broke Sousa to love in game nine.
Bedene then served for the match, which he delivered on his first opportunity to complete victory in just over two hours.
Seeded French pair Gilles Simon and Benoit Paire were knocked out by Marco Cecchinato and Kenny De Schepper respectively.
Czech number eight seed Jiri Vesely beat Norbert Gombos of Slovakia 6-4 6-0 while qualifier Attila Balazs booked a second-round showdown with top seed David Goffin.
Fifth seed Victor Estrella Burgos suffered an unexpected early exit from the Dell Technologies Hall of Fame Open in Newport after losing 6-4 6-4 to Russian Konstantin Kravchuk.
Slovakian number six seed Lukas Lacko, though, did progress as he saw off American Reilly Opelka 7-6 (9/7) 6-4.
However, seventh and eighth seeds Tennys Sandgren and Illya Marchenko were eliminated by Frank Dancevic and Bjorn Fratangelo respectively.
Australian Samuel Groth set up a second-round match against top seed John Isner when he beat fellow qualifier Austin Krajicek 6-7 (3/7) 7-6 (7/2) 6-3, having faced a match point against him before the tie-break in set two.
There were also first-round wins for Groth's compatriots Akira Santillan and Matthew Ebden, who will face Lacko in round two.
At the Swedish Open in Bastad, number six seed Fernando Verdasco faced a test as the Spaniard battled past home player Elias Ymer 4-6 6-3 6-4.
Diego Schwartzman, the seventh seed from Argentina, progressed with a 6-1 6-3 win over German Maximilian Marterer while David Ferrer, seeded eighth, saw off Federico Delbonis 6-3 6-4.
Number nine seed Horacio Zeballos saw off Nikoloz Basilashvili 6-3 6-3, while German Dustin Brown beat Brazilian Thomaz Bellucci 6-4 6-3.
Tommy Haas, though, is out after the German lost 6-7 (5/7) 6-3 6-2 to Facundo Bagnis and Thiago Monteiro progressed after opponent Dusan Lajovic retired having lost the first set 6-3. Monteiro will play Spanish second seed Albert Ramos-Vinolas in the next round.
Andrey Kuznetsov secured a second-round match against top seed Pablo Carreno-Busta when he beat German Jan-Lennard Struff 6-3 6-4.
