Best opened the scoring after 48 minutes when he rounded Cumbrians goalkeeper Adam Collin to slot home his first goal for the club since his move from Coventry City.
Chris Hughton's men doubled their advantage in the 64th minute when Ranger, who had missed a host of opportunities, scored a close-range backheel from Wayne Routledge's low cross.
Vuckic then came off the bench to score after 74 minutes to round off the win.
There was also a first senior goal for Vuckic, who gave a glimpse of his vast potential with a brilliant turn and finish from 15 yards for Newcastle’s third.
The star of the show was Routledge, who added pace to a side still trying to get into first gear, and the way he burst away from two defenders to cut back for Nile Ranger to score was inspirational.
After speculation of who will wear the captain’s armband for the uncoming season ... with suggesting it will be Kevin ... Alan Smith had the honour at Carlisle.
Carlisle took the game more seriously than Newcastle and tackles flew in ... which upset Nolan and Peter Lovenkrands.
CARLISLE UNITED (4-4-2): Collin (Caig 71), Simek (Gibson 67), M’Voto (Wood 75), Harte, Murphy, Berrett, Thirlwell (Cregg 46), Bridge-Wilkinson (Hurst 46), Taiwo (Robson 46), Curran (Zoko 46), Madine (Price 46).
NEWCASTLE UNITED (4-4-2): Harper (Krul 46), Tavernier (R Taylor 46), Coloccini (S Taylor 46) (Henderson 81), Williamson (Perch 46), Enrique (Ferguson 46), Inman (Routledge 46), Smith (Barton 46), Nolan (Guthrie 46), LuaLua (Vuckic 46) , Carroll (Best 46), Lovenkrands (Ranger 46).