The Canaries looked set to stun the home crowd when Grant Holt pounced on a horrible error from Hilario to equalise after Jose Bosingwa’s first goal for almost three years, and Drogba then appeared to be knocked unconscious in an aerial challenge with Norwich keeper John Ruddy.
But Frank Lampard blasted home from the penalty spot after Ruddy was sent off for fouling Ramires and Juan Mata scored on his debut.
Elsewhere, Everton midfielder Mikel Arteta scored an injury-time penalty as Blackburn were made to pay for missing two second-half spot-kicks of their own at Ewood Park.
Junior Hoilett saw his penalty saved by Tim Howard a minute after half-time before Mauro Formica shot against the post.
Arteta then delivered the killer blow in injury time, after Christopher Samba climbed on the back of Marouane Fellaini, to hand Everton their first win and leave Rovers pointless.
At the Liberty Stadium, Swansea’s inability to take their chances means they still await their first Premier League goal and win after drawing with Sunderland.
Record signing Danny Graham headed their best chance wide when unmarked inside the box, while the striker forced two fine saves from Simon Mignolet and Scott Sinclair stuck the bar.
Sunderland also continue to wait for their first win this term and this result will have done little to lift the pressure on boss Steve Bruce.
Earlier in the day, Franco Di Santo’s double earned Wigan a 2-0 victory at the DW Stadium as QPR new boy Joey Barton watched from the stands.
The Argentinian smashed home a right-footed effort five minutes before the break and put the result beyond doubt after 66 minutes when his deflected effort looped over Paddy Kenny.
Wolves survived a second-half battering in the day’s other early kick-off to earn a hard-fought 0-0 draw at Aston Villa.