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 television match official ruled against Greig Laidlaw - on for Parks - after 63 minutes, and for all Scotland's endeavour they could not breach the white wall of defenders as England returned home with the Calcutta Cup following a first win in Edinburgh since 2004.
England interim head coach Stuart Lancaster had named a side showing 10 changes from the last Test - the Rugby World Cup quarter-final loss to France, Martin Johnson's last as boss - and was rewarded.
The pack, led by captain Chris Robshaw, stood toe-to-toe with their more experienced rivals as the new faces - and one old in Hodgson, making his first start in four years - ensured the defending champions prevailed. Next up on the road to redemption following the ill-fated World Cup campaign are Italy next weekend.
Scotland head coach Andy Robinson, meanwhile, remains without a win over his compatriots in four attempts, with the perennial problems of numerous errors and a continual failure to find the try-line.
Robinson now travels to World Cup semi-finalists Wales with a record of two wins from 11 Six Nations matches in charge, facing the prospect of another disappointing tournament.
While for England it may be the dawning of a new era, for Scotland it was a missed opportunity.
Scotland were smarting from their last Test - the last-gasp loss to England which saw them eliminated from the Rugby World Cup at the group stage for the first time - and determined to finally realise their undoubted potential.
Robinson promised to stay true to the fast-paced, expansive game he has fostered and create "chaos".
Lock Jim Hamilton did just that in the opening minute, rampaging into the England 22 only for a misplaced pass to allow Tom Palmer to hack the danger clear.
Farrell missed a penalty from 47 metres after 13 minutes, but soon had another opportunity.
Hodgson's kick cross-field was dropped by Rory Lamont, David Strettle collected, scampered forward and fed Mauritz Botha, who was tackled and not released.
This time Farrell put England 3-0 ahead after 23 minutes.
Parks levelled three minutes later after Scotland punched forward from the restart and the visitors were penalised for going off their feet.
A Max Evans burst pushed Scotland forward before Chris Ashton was caught by Chris Cusiter, David Denton and Alasdair Strokosch, the wing failing to release, presenting Parks with another chance. The fly-half made it 6-3 after 33 minutes before his error in the opening moments of the second half allowed England to regain the lead.
Parks delayed over a straightforward clearance at his own line, Hodgson charged down and won the race for the ball. Farrell converted to make it 10-6.
The incident appeared to sap confidence from the hosts and quietened a previously vocal Murrayfield crowd.
Opportunities for Denton and Strettle fizzled out before Richie Gray powered through a gap up to the England 22, but Strokosch knocked on when following up in support.
Robinson turned to his replacements' bench with 21 minutes remaining - Alastair Kellock, John Barclay, Mike Blair and Laidlaw introduced for Hamilton, Strokosch, Cusiter and Parks.
The reinforcements had a near-immediate impact when Laidlaw appealed for a try after chasing his own chip, only to fail to get the required downward pressure ahead of Ben Youngs.
Ross Rennie darted through a gap and attempted to find Blair, only for Ben Foden to snuff out the pass.
Farrell missed a penalty attempt from halfway after 71 minutes as England spurned the chance to take a seven-point lead.
England replacement Mike Brown earned a penalty after Scotland failed to release and Farrell this time booted England further ahead with five minutes remaining.
Scotland continued to look for opportunities but were repelled on each occasion by England, who saw out victory allowing Robshaw to lift the Calcutta Cup in his second Test and first as captain.
PA
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