Two Dan Parks penalties handed Scotland a 6-3 half-time lead and put the hosts on course for just a second opening weekend win in 13 attempts in the tournament before he turned villain 24 seconds into the second half.
Charlie Hodgson charged down a Parks kick before applying the scoring touch. Farrell converted and finished with eight points on his debut.
The Saracens centre told BBC Sport: "That's just the start for us. We came out here and worked unbelievably hard for each other, that's what we've worked on the last couple of weeks - getting together and bonding as a team.
"We really want to fight for each other and we showed that out there today. We can still get better and better as the weeks go on."
Scotland flanker David Denton, who claimed the man of the match prize, added: "First of all it's very disappointing to get the loss. It was an incredible game for me to get my first start in the Six Nations but unfortunately we didn't win.
"We made a few small errors in the red zone in attack and defence and we need to work on that next week."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.