Smith helped Graham Henry guide New Zealand to World Cup glory last autumn and had initially been linked with a role with England when Nick Mallett was favourite to succeed Martin Johnson as permanent head coach.
But the appointment of Stuart Lancaster following an impressive 6 Nations showing meant the RFU were keen to make Farrell's position full time after the ex-England rugby union and league international had been employed on a temporary basis during the tournament.
"We will continue to work hard with Stuart Lancaster to build a world-class coaching team as we look towards the Rugby World Cup in 2015," said RFU chief executive Ian Ritchie.
"Andy was an important part of the coaching team in the 6 Nations and we fully respect his wishes to continue as a coach at Saracens."
Farrell sighted unfinished business as the main reason behind his decision to stay with Sarries, with whom he has another two years left on his contract.
The 36-year-old was a key part of their Aviva Premiership success last season but defeat to Clermont Auvergne in the quarter-finals of the Heineken Cup last weekend shows there is still much to work on.
"The job isn't anywhere near half done and I have decided I want to help finish the job," said Farrell.
"It has been a special privilege for me to be involved with the Saracens coaching staff for the past two and a half seasons.
"We have made decent progress during this period but, in truth, as a club, we have barely scratched the surface of our potential."
Lancaster is now expected to name an interim assistant for the summer tour of South Africa, with London Irish's Mike Catt and Clermont's Alex King among those thought to be under consideration.