Farrell put England in front with a 17th-minute penalty and Gareth Anscombe responded with one of his own before Tom Curry went over for a converted try to give the visitors a 10-3 lead at half-time.
Anscombe reduced the deficit to one point with two more penalties before Cory Hill and Josh Adams both scored tries to cap a superb comeback at the Principality Stadium.
And Farrell revealed he was unhappy with his own performance and conceded they struggled to deal with the second-half Wales onslaught.
“We didn’t really get a foothold in the last 30 minutes of the game,” he said.
“We didn’t lose our discipline. We made a few errors, myself included in that.
“We couldn’t really get out of our half. We made a few mistakes and had a few calls go against us.
“There’s a big chunk of the last part of that game where we didn’t do what we could.”
England coach Eddie Jones echoed his fly-half’s words but paid homage to Wales for their resilience.
He said: “It was one of those nick and tuck games and with small margins.
“It was a tough game. I don’t know whether we lost our heads but they put us under a lot of pressure.
“I just think a few of our players were a bit off today and that happens sometimes.
“Full credit to Wales as they deserved the victory and played very well.”
Wales have now won a best-ever 12 Test matches in a row.
And captain Alun Wyn Jones says he is hoping to extend that impressive run when they face Scotland at Murrayfield in their next Six nations picture.
“I dunno, hopefully there’s a few more ahead,” he said when asked if it was a special night for Welsh rugby.
“The people we had and the people who came on today stood up.
“Credit to England – the exchanges we had in that second half, we knew it was going to be a turnover or a bad bounce of the ball that decided the game.”