Sent in to bat at the Sydney Cricket Ground, Heat opener Josh Brown struck 53 to propel his side to a competitive total of 166-8 from 20 overs which proved too much for the Sixers, who folded for 112 all out with 15 balls remaining.
"I feel like we were the best team all year so I think we deserve to have the medal around our neck," said man-of-the-match Johnson.
"The Brisbane Heat have changed my life, it's pretty special."
Johnson and fellow paceman Michael Neser (1-15) stunted the Sixers' run-chase by accounting for openers Jack Edwards (16) and Daniel Hughes (1) in the first powerplay.
On a slow pitch, wickets tumbled during the middle overs and the Heat enthusiastically celebrated the departure of home skipper Moises Henriques, caught for 25 off the bowling of Paul Walter (1-24).
Johnson concluded his spell with a caught and bowled to send Hayden Kerr on his way for six, before Xavier Bartlett (2-12) sealed the victory with consecutive wickets in the 18th over.
Flying high after a blistering playoff-winning ton for the Heat against Adelaide Strikers, Brown survived two tough dropped catches in the deep as he hit three sixes in his fifty, ultimately out lbw to retiring spinner Steve O'Keefe (1-26).
Matt Renshaw (40 off 22 balls) and Max Bryant (29 off 19) combined for a half-century partnership to boost the visitors before seamer Sean Abbott (4-32) helped close out the innings with a triple-strike in the penultimate over.
"Another awesome year, we just couldn't get over the line tonight," said Sixers vice-captain Hughes.
The Heat, who topped the standings during the regular season, previously won the BBL championship in 2013.
