The Australian full-back kicked five in a dour first half, while Leicester's Ian Humphreys kicked three, as referee Ashley Rowden's whistle dominated.
The Falcons increased their lead after the break with another Burke penalty and a drop goal from Tom May.
Henry Tuilagi's try eight minutes from time gave Leicester hope but Burke's 76th-minute penalty sealed victory.
Leicester coach Pat Howard:
"You expect your depth to do better - we still had a good side out. "We had four British Lions out on the paddock and plenty of internationals. We should have done better.
"Earlier in the year we tried a few guys. You'd expect them to do better."
Newcastle: Burke (capt); May, Tait, Mayerhofler, Elliott; Flood, Grindal; Ward, Long, Morris; Perry, Parling; Finegan, Woods, Buist.
Replacements: Charlton for Grindal (64), Williams for Ward (71), Thompson for Long (71), Grimes for Parling (68), McCarthy for Finegan (64).
Not Used: Phillips, Walder.
Leicester: Vesty; A Tuilagi, Smith, Hipkiss, Lloyd; Humphreys, Healey (capt); Rowntree, Buckland, Morris; L Deacon, Kay; Abraham, W Johnson, Jennings.
Replacements: Dodge for Smith (58), Bemand for Humphreys (65), Taikaufa for Buckland (59), Holford for Morris (64), Cullen for Kay (55), H Tuilagi for Johnson (65).
Not Used: Skinner.
Att: 10,200
Referee: Ashley Rowden (RFU).