Giant-Shimano's Degenkolb, 25, crossed the line marginally clear of irate Frenchman Nacer Bouhanni in second, who believed the German blocked his passage during the sprint, and Dutchman Moreno Hofland in third.
Matthews finished safely inside the lead group to extend his lead to 13 seconds over Colombian Nairo Quintana, and 20 seconds over Spain's Alejandro Valverde, both of whom finished in the pack.
As the sprint finish intensified Bouhanni became boxed in between Degenkolb and the barriers and was unable to find the room to launch his final attack.
Bouhanni appealed to race officials but the German remained the stage winner.
"I don't think I changed my line," Degenkolb said. "I stayed from the beginning on the right and if he wants to pass he has to go on the left side because there was enough space."
Time trial specialist Tony Martin and Dutchman Pim Ligthart made up the day's breakaway and built up a lead of four minutes over the chasing peloton.
Martin eventually succumbed to intense temperatures and dropped back to the main field with 70km still to ride.
Ligthart powered on but his solo charge proved futile and with 30km remaining he too was caught by the peloton.
Entering the finish Bouhanni became trapped and could only gesticulate as Degenkolb crossed the finish line.
Thursday's stage six is a 167.1 km ride from Benalmadena to La Zubia.
(Reporting By Tom Hayward; editing by Toby Davis)
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