United controlled the opening quarter, Michael Carrick drawing a low stop from Courtois before the Belgian superbly clawed away Anthony Martial’s effort after cutting into the left channel.
The visitors kept up their pressure as Chelsea struggled to gain a foothold in possession, though De Gea had to react smartly to catch Nemanja Matic’s glancing header from Willian’s delivery.
Costa placed a low shot narrowly wide of the far post as Hiddink’s side suddenly began to look a real threat, with Oscar then blazing over the bar from close range after being teed up by Cesar Azpilicueta.
Wayne Rooney almost turned home Matteo Darmian’s drilled cross but John Terry had shouts for a penalty waved away in the final minute of the half as his shot on the turn took a slight deflection off Daley Blind’s raised arm.
Rooney drew a strong save from Courtois and Martial dragged a shot narrowly wide as United began the second half back on the front foot, with the Belgian again at full stretch to keep out Lingard’s curling effort.
Hiddink introduced Eden Hazard from the bench but was forced into a second substitution within a minute after Zouma had to be carried off on a stretcher after jarring his right leg when landing awkwardly near the centre circle.
Chelsea’s defence was slow to settle after Gary Cahill’s introduction and Lingard capitalised superbly to put United ahead, controlling Cameron Borthwick-Jackson’s low cross from the left and firing into Courtois’ top-left corner on the turn.
De Gea reacted brilliantly to tip Ivanovic’s volley over the bar as Chelsea sought an immediate response, and the Spain international denied compatriot Fabregas with another fine save to keep the midfielder’s effort out of the top corner.
United’s resistance was finally worn down in the first minute of stoppage time, however, as Costa capitalised on Blind’s slip to finish at the second time of asking and preserve Hiddink’s unbeaten run.