Extra-time followed, during which Branislav Ivanovic fired a rasping finish into the roof of the net after Drogba picked him out.
Leading 5-4 on aggregate, the hosts defended stoutly and could have added more as Napoli crashed out.
Chelsea, fifth in the Premier League, enter the quarter-final draw as the only remaining English side.
Di Matteo was overseeing his second game in charge after Andre Villas-Boas’s sacking, and he maintained his 100 per cent record against a free-scoring Napoli side who hit six in their previous Serie A match against Cagliari.
But Walter Mazzari’s team also conceded three in that match and Chelsea exploited their defensive fragility while largely shacking danger-man Edinson Cavani, who went missing at the Bridge.
The opening was worrying for Chelsea though, as they were repeatedly cut apart by rapid Napoli breaks.
The first chance fell to the hosts on six minutes – Daniel Sturridge’s half-volley kept out by Morgan De Sanctis – but from that point quickfire counters wreaked havoc: a Mark Hamsik effort was deflected on to the legs of Petr Cech, Cavani finished into the side netting from close range, while Lavezzi forced Cech into a save at his near post.
Just after the quarter hour the tide turned as Juan Mata saw a goal chalked off after Sturridge flicked on from an offside position.
Napoli defend with a back three, and Chelsea worked out the best way to test it: quick balls from wide but short of the byline, exploiting the space between the wingbacks and the external centre-backs.
One such ball – on 28 minutes from Ramires on the left – landed perfectly for Drogba, whose diagonal run allowed a bullet header that left De Sanctis with no chance.
With Chelsea needing a 2-0 win, it was game on, highlighted by Di Matteo’s celebrations, which saw him slip amidst the joy.
Soon afterwards Sturridge flashed a vicious low ball across the face of goal, but via deflection it flashed inches wide of both Drogba’s lunge and the far post.
Mata then put in a superb cross from the right that Sturridge flicked on, but the England forward’s intervention took it away from an unmarked Drogba; yet another ball from wide – from a surprisingly solid David Luiz – was somehow kept from being tapped in by Mata and Sturridge thanks to a sliding intervention by Paolo Cannavaro.
Chelsea were ascendant and they got the perfect start to the second half: Napoli conceded a sloppy corner, Lampard’s delivery met by Terry, who was pinpoint accurate with his angled header into the top left.
The defending was sloppy – Gokhan Inler should not be marking the main aerial threat – but Chelsea went ahead on away goals.
Napoli responded brilliantly, pegging Chelsea back to their final third and denied by some good defending from Luiz and Terry.
Their away goal was thus deserved – and easy on the eye too: a clearance landed on the chest of Inler, who controlled expertly before rattling a ferocious half-volley from 20 yards into the bottom left, Cech barely able to get down. It was 2-1, and advantage Naples.
As Napoli had responded well, so did Chelsea, who went close through Ivanovic and Drogba, denied by De Sanctis acrobatics.
The visitors still threatened – Juan Camilo Zuniga exploited an Ashley Cole slip but his shot hit Cech and an unaware Lavezzi – but the latter stages were all about Chelsea, who pounded the Italian goal.
Terry had just missed Drogba’s flick at a corner when, as Ivanovic rose to meet another set-piece delivery, Lavezzi – who had been the visitors’best player – stuck out an arm.
It was a clear penalty, which Lampard drilled home with power, and with 15 minutes left the match was poised at 4-4 on aggregate.
Before the end of normal time Napoli carved out a great opportunity, Lavezzi heading inches wide after escaping Luiz to meet Hamsik’s cross.
In injury time there was controversy when Drogba was hauled down by Salvatore Aronica as the Ivorian prepared to head Ramires’s centre in, the referee and additional officials missing the double-handed tug from the Napoli defender.
In extra time the fresh legs of substitute Fernando Torres caused problems for a tiring Napoli, the striker impressing down the right as he won a corner which Ivanovic headed wide; Napoli went down the other end and almost scored, Hamsik’s volley from a Lavezzi cross beating Cech but also the woodwork.
Torres managed to miss his now obligatory chance, putting wide under pressure from Hugo Campagnaro after De Sanctis went walkabout, now 24 hours without a goal.
But it mattered not as – after a Lampard free-kick was not fully cleared – Drogba found a low cross that Ivanovic drilled into the roof of the net.
It was the final act of the first half of extra-time and, as they switched ends for the remainder, there was only one winner.
Napoli were spent and Chelsea should have added more – Ramires passing too early to Torres when the right ball would have sent him through, Drogba volleying well wide after good work from Florent Malouda caught the guests on the break.
The final whistle soon went and Di Matteo was joined by players and staff on the pitch, Terry – who had come off late on – leading the celebrations as the Blues made it through.