Swedberg got his chance from the start of the second half when he came on in place of Pablo Duran and with the score tied at 0–0.
Seven minutes later, the U21 international headed in Celta's sensational opening goal with his heel – a goal that will make headlines.
But he wasn't ready for the evening. In added time, he dribbled – almost walking – past Thibaut Courtois and rolled in 2-0.
Swedberg has had an injury-filled season, missing eight games in October and November and has just returned.
After the dream goal, the striker was heavily criticised by increasingly frustrated Real players. Fran Garcia was shown a red card for a foul with just under half an hour left to play.
Central to Celta's back three, centre-back Carl Starfelt impressed in a defence that managed to keep world stars like Kylian Mbappé, Vinicius Júnior and Jude Bellingham scoreless.
Celta Vigo are in mid-table. For Real Madrid, the loss means the club is now four points behind leaders Barcelona. Disciplinary action is also expected. Alvaro Carreras received Real's second red card at the end, with several players expressing their dissatisfaction with the referee.
When Barcelona scored 5-3 at Real Betis on Saturday, national team player Roony Bardghji celebrated his first league goal for the club.
