The Brazilian winger netted his first Champions League goal on the 37th-minute mark to give the visitors a half-time lead at the Parc des Princes, but PSG quickly turned the game on its head with two strikes in the first five minutes of the second half from ex-Barca man Ousmane Dembele and Vitinha.
However, Raphinha restored parity for the Catalan giants with a superb volley midway through the second half before substitute Andreas Christensen scored 13 minutes from time, just seconds after coming off the bench, to give Xavi's side a slender advantage heading into next week's second leg in Spain.
Neither Christensen or Sergi Roberto will be available for next Tuesday's encounter with PSG, though, as both players will serve a one-match ban due to an accumulation of yellow cards.
Luis Enrique's men made a bright start to the contest, with Kylian Mbappe - who has a surprisingly quiet night - and Marco Asensio both striking at goal in the opening 10 minutes, but Barca gradually grew into the game and took control of proceedings.
Nuno Mendes spared Gianluigi Donnarumma's blushes with an instinctive goalline clearance with his weaker right foot after the PSG goalkeeper failed to collect the ball from a corner after being beaten in the air by the towering Robert Lewandowski.
Donnarumma then made a brilliant one-handed save low down to his left to tip a powerful 25-yard strike from Raphinha around the post, but the Italian did not cover himself in glory eight minutes before the break.
Lewandowski, and standout performer for Barca in the first half, began a counter-attack and played through Lamine Yamal on the right flank, before the 16-year-old looked to return the favour by playing a teasing outside-of-the-boot pass towards his Polish teammate running into the box.
Donnarumma sprinted off his line to intercept, but he could only tip the ball straight into the path of Raphinha, who initially slipped before composing himself to slot a first-time right-footed shot high into the net.
PSG ended the first 45 second best, but Enrique's half-time team talk certainly did the trick and they miraculously turned the game on its head within the first five minutes of the second half.
The hosts exploded out of the blocks and restored parity on the 48-minute mark when Dembele pounced on a loose clearance inside the box from Ronald Araujo and dropped of the shoulder to fool Frenkie de Jong before unleashing a powerful left-footed shot that flashed beyond a hapless Marc-Andre ter Stegen.
Just two minutes later, PSG were ahead following a well-worked passing sequence that resulted in Fabian Ruiz playing a perfectly-weighted through-ball down the right for Vitinha, who poked a first-time finish under the outstretched arm of Ter Stegan and into the bottom corner.
Barca were left stunned and they had their backs against the wall as PSG searched for a third; Bradley Barcola - who replaced Asensio at half time - came close to increasing the hosts' lead just before he hour mark, but his shot from a tight angle was saved onto the crossbar by Ter Stegen.
Xavi brought on both Joao Felix and Pedri for the final half-hour in search of a much-needed spark in the final third, and the latter made an instant impact as he set up Raphinha for his second goal of the night, and eighth of the season, to level the scores.
Just a minute after entering the field of play, Pedri received the ball in a central position from around 35 yards out and he floated a delightful ball over the top of PSG's backline towards Raphinha, whose exquisitely controlled left-footed volley with the outside of his boot flew past Donnarumma and into the bottom corner.
In search of a hat-trick, Raphinha then bent a 25-yard free kick around a PSG wall, but it was denied by Donnarumma low down to his right, before Dembele at the other end opened up his body and placed a right-footed shot against the outside of the post after receiving a brilliant pass on the left from Vitinha.
With 15 minutes remaining Xavi made another double substitute, calling upon both Ferran Torres and Christensen, and although the latter was not brought on in an attacking sense, the defender found himself in the right place at the right time to put Barca in front just seconds after his introduction.
Ilkay Gundogan's dangerous inswinging corner was met by Christensen from around six yards out, and with his first touch placed a cushioned header into the centre of the goal out of Donnarumma's reach.
PSG pushed for a late leveller in the closing stages, but Barca restricted to the hosts to minimal clear-cut chances and held on for a deserved victory in Paris.
Enrique's men are not in domestic action this weekend, so they will shift all their attention to next Tuesday's second leg in Spain, while Barca travel to Cadiz for a La Liga fixture on Saturday.