Following a closely-contested 1-1 draw in the first leg in Italy three weeks ago, Barca came out on top by a 3-1 scoreline in the second leg at the Estadi Olimpic Lluis Companys on Tuesday night.
The Catalan giants made an electric start to the rip-roaring contest with Fermin Lopez and Joao Cancelo scoring just two minutes apart in the opening 17 minutes, before Napoli pulled one back through Amir Rrahmani on the half-hour mark.
Napoli improved after the break and both side had their fair share of chances, but not long after Jesper Lindstrom missed a sitter for the visitors, Robert Lewandowski - who has a quiet night in front of goal - sealed the victory for Barca with a simple finish seven minutes from time.
Xavi has successfully steered Barcelona into their 19th Champions League quarter-final - only Real Madrid (22) and Bayern Munich (20) have reached the same stage on more occasions - and the five-time winners will learn their opponents when the last-eight draw is made on Friday.
As for Napoli, their wait to beat the Catalan outfit goes on, as they have now lost three and drawn three of their six meetings; the Italians have also become the fourth Serie A champion in the last five seasons to be knocked out in the Champions League last 16.
Xavi made three changes to the Barcelona side that beat Mallorca 1-0 in La Liga last weekend, with Lewandowski, Lopez and Ronald Araujo all coming into the team, while 17-year-old centre-back Pau Cubarsi was handed his Champions League debut - becoming the youngest-ever defender to start a knockout match in the competition.
Meanwhile, Napoli head coach Francesco Calzona opted to make just two alterations to the lineup that began a 1-1 draw with Torino in Serie A last time out, with Rrahmani returning in central defence and Bournemouth loanee Hamed Traore starting in midfield.
Victor Osimhen led the line for Napoli and was presented with a glorious chance to open the scoring just four minutes in when he met a ball over the top to run in behind Barca's backline, but the Nigerian never had the ball under complete control and could only poke a tame effort towards Marc-Andre ter Stegen - Osimhen would have been ruled offside anyway.
Barcelona dominated proceedings after that, though, and were full of confidence as they burst forward on multiple dangerous attacks into the final third. Lopez drilled a shot wide not long after running in behind and lifting a shot over the crossbar.
However, the 20-year-old midfielder made no mistake at the third time of asking, as he opened the scoring in the 15th minute with a composed side-footed finish into the bottom-left corner - his fifth goal of the season and second in the Champions League - after meeting a clever cut-back from Raphinha on the left.
Within a flash, Barca doubled their advantage on the counter-attack, with livewire Lamine Yamal setting up Raphinha to drive forward before smacking the post with a venomous right-footed shot, but the rebound fell kindly to Cancelo to slot home a calm first-time finish.
Barcelona were deservedly in front, while Napoli were far from convincing in possession and lacked intensity, but out of nowhere, they managed to pull one back on the half-hour mark; Matteo Politano made a smart run down the right before pulling back a low cross to centre-back Rrahmani, who continued his run into the danger zone to place a striker's finish into the bottom corner.
Both teams cancelled each other out for the rest of the first 45, but Napoli made a much brighter and confident start to the second half, with Khvicha Kvaratskhelia whipping a 25-yard shot narrowly wide of the post before Politano skipped past Cancelo with ease and had a powerful strike blocked by Cubarsi.
Napoli enjoyed an extended period in possession, but Barcelona reminded the visitors of their threat when Cancelo ran down the left and picked out Raphinha for a side-footed shot inside the area, which was beaten away in awkward fashion by Alex Meret.
In a bid to regain control of the contest, Barca boss Xavi opted to bring on two experienced midfielders on the hour mark in Oriol Romeu and Sergi Roberto, with Lopez and Andreas Christensen both making way.
Neither Romeu or Roberto were able to make an immediate impact as Napoli raced forward on another promising attack, this time involving Stanislav Lobotka, whose looped cross was carelessly headed back into the danger zone by Cancelo, allowing Osimhen to tee up Andre-Frank Zambo Anguissa, but his power-drive was blocked.
Barca thought they had scored a third in the 68th minute when Lewandowski had a towering header from a corner parried by Meret, before the ball was drilled back into the six-yard area for Yamal to tap home, but the 16-year-old was ruled offside.
Both teams then came agonisingly close to scoring, Yamal firstly received a cushioned pass from Ilkay Gundogan to curl a left-footed shot just inches wide of the far post, before Napoli substitute Lindstrom somehow directed a free header wide at the back post - a huge opportunity that went begging for the visitors.
That missed chance proved costly as Barca put the game to bed in the 83rd minute following a superb passage of play, with Gundogan feeding Roberto inside the area to square it to Lewandowski for a simple tap-in.
Napoli substitute Mathias Olivera fired a cross-cum-shot against the crossbar from a tight angle, shortly before Kvaratskhelia almost skimmed the post with low-driven shot in the closing stages, but the Italians were left frustrated as Barca secured safe passage through to the last eight.
Barcelona now turn their attention to Sunday's La Liga battle with Atletico Madrid at the Wanda Metropolitano, while Napoli make a daunting trip to San Siro to lock horns with high-flying Serie A leaders Inter Milan.