Giant-Shimano rider Degenkolb held off the challenge of Spain's Vicente Reynes and Matthews, who finished third, in a sprint finish.
Colombian Nairo Qunitana is second overall, eight seconds behind Matthews, with Spain's Alejandro Valverde a further seven seconds adrift in third.
Twice Tour de France yellow jersey winner Contador, and 2013 champion Chris Froome, who both crashed out of the this year's Tour, sit 27 and 35 seconds behind Matthews respectively.
Degenkolb told reporters: "Yesterday was disappointing not to hang on in a hard and explosive finish and I always need one or two days to adapt to the heat.
"But today I felt much better. My team did a great job to get me there and it's always a good feeling to win a stage on the Vuelta."
A host of prominent sprinters struggled on a particularly hot day in southern Spain but Degenkolb was excellently protected by his team mates.
Valverde was among a group of three riders who attacked near the summit of the day's final climb with about 25km remaining.
They were eventually caught with 10km left before Degenkolb powered away in the closing straight to win by a surprisingly comfortable margin.
Wednesday's stage five is a 180km ride from Priego do Cordoba to Ronda.
(Writing by Tom Hayward; Editing by Ken Ferris)
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