Eddie Jones's side triumphed 22-19 after extra time thanks to a sudden-death penalty from Farrell against an inexperienced France side missing 25 senior players.
Les Bleus led by seven points with four minutes of normal time remaining, but a last-gasp try from Luke Cowan-Dickie and a Farrell conversion prolonged a contest that was eventually settled by the captain's boot.
It was a difficult outing for Farrell, who missed four penalty attempts – the most he ever has in a Test match – including one effort that struck the post early in extra time.
However, the Saracens star held his nerve at the critical moments to seal England's eighth consecutive Twickenham victory over France and become only the sixth man to record 1,000 points in Test rugby.
He told Amazon Prime Sport: "We probably didn't come out of the blocks in the first half but the performance when we were on the back foot, and to keep giving me chances to win the game... it took a while! I thought the boys were excellent.
"To be fair, they gave me a few chances of winning it and I didn't quite make them. I was just thankful we got the win. The boys deserved it after that second-half performance.
"Of course there will be stuff to work on, especially in that first half, and of course there's always stuff to get better at and that will be highlighted.
"But I thought the attitude we came out with in the second half, and when we were under pressure a bit towards the end to get that try and take it into extra time, and to get us two shots at goal to get us two chances to win it, I thought it was a great effort.
"It's been a good nine weeks together. We feel like we've grown again and it's not too long before we get back together again."
England's man of the match Billy Vunipola praised France for a clinical and disciplined performance in which they converted all five penalty kicks and won 10 turnovers to England's four, missing only 20 of 149 tackles.
"They were outstanding, they challenged us all over the pitch," Vunipola said. "They are a great team and we are lucky we came out on the right side of the result.
"Eddie reassured us we can do this. It is another trophy and we keep improving, trying to be the greatest team in the world."