After Neymar's 18th-minute strike had given the Catalan giants an important away goal at the Parc des Princes, Suarez scored twice in 12 second-half minutes to help Barca put one foot in the last four.
Having nutmegged substitute David Luiz, the Uruguayan held off Marquinhos and Maxwell to nudge home his first, before outfoxing Luiz again en route to a beautifully-finished strike.
That set the seal on the result and put Barca on the verge of a seventh European semi-final in eight seasons, with Gregory van der Wiel's deflected strike eight minutes from time doing little to offer Laurent Blanc's side hope.
Having fallen behind early on, the French champions were also hit by the injury-enforced withdrawal of captain Thiago Silva - and that loss was keenly felt.
His absence added to the omissions of energetic midfielder Marco Verratti and talismanic striker Zlatan Ibrahimovic - both serving one-match bans - and the hosts now face an uphill battle in the return leg next Tuesday.
Following a low-key start, Barca - for whom Dani Alves was suspended - gradually got into their groove and went ahead when Lionel Messi struck Salvatore Sirigu's right-hand post with a curled strike 13 minutes in.
The visitors' persistence was eventually rewarded when Neymar opened the scoring five minutes later - Sergio Busquets dispossessing Adrien Rabiot and allowing Messi an opportunity to craft a perfect throughball.
Neymar made no mistake with his finish and PSG were dealt a double blow in the aftermath of the goal, as captain Silva was forced off to be replaced by Luiz.
Javier Mascherano - one of three Barca changes from the weekend Liga draw with Sevilla - produced a last-ditch tackle to deny Edinson Cavani a sight on goal as Blanc's men enjoyed their first spell of sustained pressure.
Despite ending the half strongly, PSG almost fell further behind prior to the break when Suarez dispossessed Luiz, only to shoot tamely at Sirigu.
The hosts' momentum continued into the second half as long-range strikes from Javier Pastore and Rabiot caused problems Marc-Andre ter Stegen.
However, Suarez took the game beyond PSG with a sublime first after 67 minutes, as he showed skill and strength to beat three markers and prod past Sirigu - registering Barca's 400th Champions League goal.
Suarez cleverly evaded Luiz again and made it three soon afterwards with the only blow to their night coming when Jeremy Mathieu diverted Van der Wiel's strike past Ter Stegen with time running out.
Mathieu made amends with a timely block from Cavani's strike before the end, helping maintain Barca's two-goal lead ahead of the second leg at Camp Nou.
