Farrell finished the night with 12 points from four penalty kicks, while Elliott Daly also converted a huge effort from his own half for the Lions just after half-time.
In 129 years of tours to New Zealand, Australia and South Africa, it was the Lions' first drawn series against New Zealand and only the second in their history after they finished 2-2 against South Africa in 1955.
Lions captain Sam Warburton, who lifted the series trophy with his opposite number Kieran Read, said: "I'm not sure what to make of that really. I was getting ready for extra time."
After a vibrant opening 10 minutes, Daly beat Israel Dagg in the air to take the Lions up to the New Zealand line, only for the two rookie All Blacks to turn defence into points.
An intercept by Jordie Barrett set Ngani Laumape racing 80 metres before he was cut down from behind by Jonathan Davies.
From a resulting scrum, Beauden Barrett kicked across field for his brother, Jordie, to leap high and instinctively palm the ball back to Laumape for the opening try which Beauden Barrett converted to make it 7-0.
The Lions replied with two penalties from Farrell to reduce the arrears to 7-6.
Just before half-time, Brodie Retallick won a lineout and the All Blacks went wide for Jordie Barrett to score on the right and give the hosts a half-time lead of 12-6.
A third penalty to the Lions, this time a 52-metre shot from Daly, lifted the Lions at the start of the second half and Farrell's third successful penalty put them level with 20 minutes remaining.
In the final minute, the All Blacks won a penalty within range of the goalposts for a Ken Owens offside, only for referee Romain Poite to rule that the offence was accidental and reduce the punishment to a scrum.
"Maybe it's the right result for the series. Both teams played well in patches and we just couldn't be split," said All Blacks skipper Read, who was playing his 100th Test.
"It's just one of those ones. I don't know what to feel, it's a funny one."
All Blacks head coach Steve Hansen added: "It's come down to the wire and we've ended up with one hand on the trophy each which is a bit like kissing your sister. As little kids we are told to take the good with the bad and we have to do that," he added, referring to Poite’s decision in the dying seconds.
"It's a decision the ref has made and we will live with it."