Marko Arnautovic scored the only goal of the game as Stoke inflicted a seventh defeat of the league season so far on Chelsea, winning 1-0 at the Britannia Stadium on Saturday in the absence of the visitors' banned manager Jose Mourinho.
Costa was kept on the fringes of the action by the home side's defence and Shawcross was delighted to have triumphed in his personal battle with the Brazil-born Spain international.
"Diego Costa gets in your face, he wants to bully you so you have to stand up to him," he told Sky Sports. "He is a fantastic player who plays close to the line, so to keep him quiet is fantastic.
"We have a fantastic spirit and a good team now. Look how many Champions League winners' medals we've got in the squad now."
Costa was caught up in controversy after the match after being reported to referee Anthony Taylor following an off-the-pitch incident with a Britannia Stadium steward.
The 27-year-old appeared to stand on the steward's foot after retrieving the ball for a throw-in, though a Chelsea spokesperson described the incident as "utter nonsense".