While some feared the Cottagers may struggle with a hangover from their midweek European exertions, they impressed from the offset and dominated throughout against their neighbours from less than three miles up the road.
Johnson handed the home side the lead after just 79 seconds and their strong start continued when Danny Murphy slotted home from the spot midway through the first period.
QPR looked poor in the rivals' first meeting since 2001 and hopes of securing their first win over Fulham since 1983 were all-but curtailed just before the break when Johnson tucked away his second of the afternoon.
The 30-year-old ensured he would leave Craven Cottage with the matchball in the 60th minute when latching onto Murphy's quick free-kick, before Clint Dempsey and Bobby Zamora wrapped up the 6-0 rout.
The London derby started at ferocious pace, with Shaun Wright-Phillips' curled 20-yard effort forcing Mark Schwarzer to save within the opening minute.
Fulham immediately launched an attack at the other end and broke the deadlock.
Moussa Dembele played a neat one-two with Zamora before unleashing a fierce drive that Paddy Kenny could only parry onto Johnson, who finished with ease.
The rapid start to the match continued and the Cottagers came close to doubling their lead immediately afterwards as Fitz Hall almost flicked into his own net from Dempsey's squared ball.
The United States international then saw a goalbound effort blocked by a last-gasp challenge by Bradley Orr and, as the home side's onslaught continued, Johnson came agonisingly close to netting his second of the afternoon after firing a volley inches wide from a Zamora header in the 17th minute.
However, Fulham did not have to wait long to score their second as Kenny upended Johnson in the box and referee Andre Marriner awarded a 20th-minute penalty, which skipper Murphy duly converted.
The Hoops were struggling to cope with the home side's attack-minded philosophy and unable to turn their fleeting attacks into goal-scoring opportunities.
Fulham, meanwhile, showed no signs of a post-Europa League hangover and looked impressive in attack, with Steve Sidwell coming close in the 37th minute.
And just one minute later, Jol's side grabbed a third of the afternoon as Johnson turned home Zamora's cross after keeping a cool a head during a melee in the box.
QPR looked shell-shocked as they trudged off at the break and came out with two new faces as DJ Campbell and Tommy Smith replaced Adel Taarabt and Shaun Derry.
Warnock's attempts to gee up his side looked to be working at the start of the second period, with both Wright-Phillips and Alejandro Faurlin getting away shots.
However, Fulham were soon back in their stride and Kenny was forced to acrobatically save a Zamora effort in the 58th minute.
Sidwell had another attempt from the resulting corner and moments later Johnson ensured he would leave Craven Cottage with the matchball when he latched onto Murphy's quick free-kick.
Fulham did not rest on their laurels, though, and after Zamora fired just wide, the striker laid the ball off to Dempsey to make it 5-0 with an exquisite finish.
QPR's first top-flight match against Fulham got even worse in the 74th minute when Zamora was rewarded for an industrious performance with a deserved goal, slotting home past a hapless Kenny.
The only moment of joy for 3,000-odd visiting fans came just before Zamora's goal as they welcomed back striker Jamie Mackie, who came off the bench for the first time since suffering a double leg break in an FA Cup tie with Blackburn in January.
The former Plymouth striker skewed a shot wide as the game came to a close, while at the other end fellow substitute Bryan Ruiz saw an angle drive fly just wide before Marriner called time.