Carter, who was hampered by a slight calf injury, was replaced early in the second half.
Johan Goosen scored all of Racing 92's points by kicking three penalties at the Lyon stadium.
Pre-match showers made the pitch slippery and both teams found it difficult to keep the ball.
With Carter injured, the kicking duties were initially handed to Maxime Machenaud, who missed a 40-meter kick in the fourth minute after Racing was awarded a penalty at the scrum.
Farrell ignited Saracens' first move in the seventh minute with a chip to chase but Brice Dulin got to the ball first to thwart the attack.
Dulin was sanctioned for deliberately knocking the ball into touch before Farrell made it 3-0 for Saracens after prop Ben Tameifuna was penalized for losing his bind.Racing's riposte came amid heavy rain.
After an unsuccessful attempt from former All Black Joe Rokocoko to break through Saracens' midfield, Goosen put the French side level in the 17th minute with a 55-meter penalty.Racing could not build on the momentum when Rokocoko failed to carry on after intercepting a pass.
Farrell missed a drop goal in the 21st minute but made it 6-3 after prop Eddy Ben Arrous conceded another penalty on the ground.Racing took a blow when Machenaud was substituted near the half-hour after a knock on the head and Saracens increased its lead through another Farrell effort, this time right in front of the posts, after Racing was caught offside.
Goosen made sure Racing stayed in the game with four minutes left in the first half, reducing the English side's lead to 9-6 but Farrell added three more points from 45 meters and Saracens returned to the dressing room with a six-point lead.
Carter, who had no impact on the game, was replaced by Remi Tales at the start of the second half and watched from the stands as England scrumhalf Richard Wigglesworth continued to apply pressure with his fine kicks into the corners.
Farrell made it 15-6 in the 51th after Dimitri Szarzewski was caught offside in a ruck before the Sarries contained Racing's assault in the last 20 minutes.
"They got a good bit of momentum heading into the final 20 minutes," Farrell said.
"But some of our guys who like the turnovers and the big hits managed to turn the tide. Everyone here is working hard to get better. Everyone makes the most of the day in front of them."
Goosen narrowed the score to 15-9 but his teammates rushed it too much in the decisive zone to breach the sturdy Saracens defense while Farrell stayed perfect until the end, ensuring Saracens became England's first European champions since Wasps nine years ago.