A dull game burst into life in the second half amid a flurry of sin-binnings for both sides, with one for England's Freddie Steward leading to a penalty try for Wales.
England were reduced to 12 men with further cards for Ellis Genge and Farrell - later upgraded to red - and Tomos Williams put them further ahead with a breakaway try.
However, Maro Itoje hit back for battling England, replacement George Ford converted and then kicked the decisive penalty four minutes from the end.
England's fans were left somewhat bewildered as they had spent a frustrating hour wondering just how their team were going to score a try and play some rugby but left delighted by the gutsy comeback.
"The last seven weeks is proof of what we have done and then we worked right to the end," said impressive flanker Ben Earl.
"We went down to 12 men at one point and we came back so we are really pleased. We showed a bit of fight after the questions been asked of us after last week."
England edged a largely forgettable first half via two Farrell penalties but neither side seemed able to build any sort of sustained attacking play.
The most notable action came after 32 minutes when England lost scrumhalf Jack van Poortvliet, who limped off with a leg injury, just as winger Henry Arundell was sin-binned for an offside tackle on Liam Williams.
Wales also lost their captain and Dewi Lake, with the hooker looking distraught on the sidelines after being helped off the pitch.
The visitors had flanker Tommy Reffell yellow-carded for killing the ball at the start of the second half, Farrell punishing Wales further from the resulting penalty.
Wales got on the board with an Owen Williams penalty but then suffered another injury blow when exciting new Number Eight Taine Plumtree went off clutching his wrist.
The game then exploded as Steward took out Josh Adams in the air. A penalty try was awarded and Steward joined Ellis Genge in the sin bin to leave England trailing 10-9 and down to 13.
It got even worse for England three minutes later when Farrell joined them after his shoulder crashed into the jaw of Taine Basham.
Wales took immediate advantage with a 70-metre break finished off by Williams.
Dan Biggar converted for a 17-9 lead but, summoning the spirit of the “White Orcs on Steroids” when they famously won when down to 13 against New Zealand in Wellington in 2003, the 12 men hit back, with seemingly all of them shoving Itoje over.
Ford converted to cut the deficit to one point and suddenly the crowd were roaring.
Joe Marchant was inches away from catching a Ford kick to the corner but Adam Beard was then yellow carded to make it 14 v 14, and Ford slotted the resulting penalty to secure the win.
If it wasn’t the flowing, dominant display England coach Steve Borthwick might have hoped for, he will be delighted with the spirit shown in adversity.
Farrell will face a disciplinary hearing later this week and is almost certain to be banned from at least next week’s game against Ireland in Dublin and against Fiji at Twickenham on Aug. 26.
While England named their World Cup squad earlier this week, Warren Gatland will unveil his after Wales’s one remaining warm up game against South Africa next week.