United started well with
Chelsea players felt they should have been given a first half penalty when John Terry goal bound shot looked like it has struck the hands of Daley Blind in the 8-yard box.
Referee Michael Oliver waved it away.
A corner from Willian was met by Nemanja Matic but David De Gea did well to save before a Costa effort flew across the face of goal.
Cesar Azpilicueta found Oscar in a good position but the Brazilian midfielder could not direct his shot on goal.
United started brighter in the second half and got a deserved lead through Jesse Lingard in the 61st minute.
Juan Mata fed the impressive Borthwick-Jackson on the left and his cross was met by Wayne Rooney before Lingard pounced and twisted to plant his effort into the corner of the net.
Chelsea had suffered a set-back some minutes before the goal when Kurt Zouma jumped and landed on his right leg with his fully body weight passing through his knee.
The way he screamed for medical held showed that it was very painful.
United were on their way to the three points before Cesc Fabregas found Costa unmarked in the box and the striker calmly beat De Gea to grab an equaliser for Chelsea.