Portugal's world number 50 was seeded eighth for the ATP 250 event in Sweden, but went down 7-6 (7-4) 4-6 6-4 to Romanian qualifier Marius Copil.
Despite sitting 131 places below Sousa in the rankings, Copil recorded victory on the back of a fine serving display featuring 18 aces.
The other seeds in action on Tuesday - Leonardo Mayer, Fernando Verdasco and Jeremy Chardy - all made it through.
Mayer and Chardy were victorious in straight sets against Donald Young and Elias Ymer respectively.
Ymer, an 18-year-old from Sweden, has won only two of his eight matches on the ATP Tour since debuting last season and had no answer to the sixth seed from France, who converted all three of the break points he created to go through 6-3 6-4.
Mayer also progressed with a degree of comfort, breaking twice on his way to a 6-4 6-4 win over Donald Young.
Verdasco was made to scrap for his place in the second round, but eventually overcame Marinko Matosevic 6-3 3-6 6-3.
The Spanish seventh seed saved all seven of the break points he faced in the first set.
In the day's other matches, Adrian Mannarino and German qualifier Matthias Bachinger overcame Marcos Baghdatis and Igor Sijsling in three-set encounters.