Saracens are the first English club to win Europe's premium club competition since Wasps triumphed in 2007.
"We have had big disappointments but each time we have come back stronger and we are happy to have finally done it," said man of the match Maro Itoje.
"We knew this was a game we could win, we had the form coming into the game, we knew if we do what we do, dominate the gameline, then we could win. We will enjoy tonight and build on from there."
"They were just a bit better in every department of the game and Carter did not have a good day, with a calf injury," said Racing president Jacky Lorenzetti.
Saracens went ahead when Farrell kicked his first penalty after Racing were penalised at the scrum, only for the French to equalise as Goosen's long-range penalty provided a timely reminder to the Sarries to stay disciplined.
Farrell added a couple of penalties in the 32nd and 36th minutes as Racing looked disorganised once Machenaud suffered a concussion in the 22nd minute and had to be substituted by Mike Phillips.
Goosen and Farrell traded penalties as Saracens, who looked firmly in control, reached halftime with a six-point lead.
Carter looked well below his usual brilliance and he was substituted by French international Remi Tales in the 43rd minute, meaning Racing were without their halfback pair and the duo of Phillips and Tales were too inexperienced to counter Saracens.
Farrell continued to kick penalties and his team's aggressive and well-organised defence kept Racing at bay.
Racing earned a lifeline after an hour when Goosen's penalty meant they needed a converted try to win their first European title.
But they were too tired for the challenge and Farrell added two further penalties to put the result beyond doubt.
(Writing by Julien Pretot; Editing by Ken Ferris and Pritha Sarkar)
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