The Canaries looked set to stun Stamford Bridge on Saturday when Grant Holt pounced on a horrible error from Hilario to equalise and Drogba appeared to be knocked unconscious in an aerial challenge with goalkeeper John Ruddy.
But Frank Lampard blasted home from the penalty spot after Ruddy was sent off for bringing down Ramires and Juan Mata scored on his debut, adding to Jose Bosingwa's first goal for almost three years.
Chelsea boss Andre Villas-Boas and captain John Terry used their programme notes to urge the home fans to get fully behind their team today after some booed them off at half-time against West Brom last week.
But supporters could be forgiven for wanting to see a performance to give them hope Chelsea have what it takes to topple Manchester United as champions this season, something called into question by their opening two displays.
Villas-Boas responded by pairing Fernando Torres and Drogba together in what was a 4-4-2 for the first time this season.
Possibly expecting a 4-3-3, Norwich began the game with a five-man defence and they duly fell behind inside six minutes.
After allowing Ramires to test Ruddy from 25 yards, they made the same mistake with Bosingwa, who brilliantly found the back of the net via the inside of the post.
Norwich began to settle and should have levelled in the 17th minute when Holt beat the offside trap but was unable to get a touch on Wes Hoolahan's brilliant ball over the top.
Chelsea's passing was becoming ragged, although Torres was close to teeing up Drogba as the pair began to demonstrate an understanding that had been lacking in previous outings.
Chris Martin wasted another great chance to equalise, Hoolahan playing him clean through only for Hilario to save before crowding out the forward's rebound attempt and tipping over Kyle Naughton's 25-yarder.
He also smothered Holt's header before City boss Paul Lambert withdrew Zak Whitbread for Anthony Pilkington on the half-hour mark and went 4-4-2.
Drogba went close with a free-kick but it was another forgettable first half for Chelsea, who might have doubled their lead within eight minutes of the restart had Pilkington not nodded Terry's header off the line or the recalled Branislav Ivanovic powered Lampard's corner wide.
Hoolahan was withdrawn for Steve Morison on the hour mark and the change took less than three minutes to pay off thanks to a moment of madness from Hilario, who needlessly came racing off his line to claim a cross and collided with Ivanovic, allowing Holt to hook the ball into the empty net.
It got worse for Chelsea as they tried to regain the lead, Drogba appearing to be knocked unconscious after an aerial challenge with Ruddy.
The striker landed in sickening fashion and medics raced onto the field to treat the stricken 33-year-old, with the game stopped for seven minutes while he was loaded onto a stretcher and carried off wearing what appeared a breathing aid.
Villas-Boas also withdrew the recalled Florent Malouda and brought on Nicolas Anelka and Mata for his debut.
Norwich showed no sympathy and Morison was put clean through, Terry thwarting him with a desperate tackle.
Holt also mistimed a free header from the resulting corner before Mata and Torres combined for the first time, the latter nodding his fellow Spaniard's cross too close to Ruddy.
It looked anyone's game but all that changed with 10 minutes remaining when a brilliant Chelsea breakaway brought a penalty and a red card for Ruddy.
Anelka sent Ramires racing clear and he was brought down by the onrushing goalkeeper.
Lambert appeared to be furious with the Chelsea bench following the dismissal and Lampard blasted the penalty straight down the middle past newcomer Declan Rudd.
Romelu Lukaku came on for his debut for Torres, sending a diving header wide and missing an Anelka cross after Norwich almost levelled again when Morison hooked over.
Ivanovic somehow failed to kill the contest when he headed wide from three yards and Lampard also missed on the rebound after Lukaku's shot was saved in the 11 minutes of stoppage-time that followed.
A third goal finally arrived with seconds to spare, Mata's debut ending in dream fashion as Ritchie De Laet presented him with the ball and he slotted beyond Rudd.