Trailing 1-0 at the Nou Camp with 10 minutes to go, Sergi poked in a rebound from close range, and moments later Suarez produced a fine finish for the winner, sweeping home first-time from just inside the area.
Kyriakos Papadopoulos had put the German side ahead after 22 minutes from a corner, with Barca keeper Marc-Andre Ter Stegen beaten in the air.
For Barca, who were without injured talisman Lionel Messi, it was a frustrating game of missed chances until their pressure told.
"We believed in ourselves until the end," Suarez told reporters.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"We are the champions. We were not able to play our best football but we had to win this match and show that we are the champions.
"The coach (Luis Enrique) gave us good advice in the break and now it is important for us to be leaders."
Barca are top of Group E with four points from two games. Leverkusen are second with three, ahead of BATE Borisov on goal difference.
Barcelona coach Enrique decided against a tactical change in the absence of Messi, with youth product Sandro Ramirez coming into their attacking trident.
Leverkusen, who were on the wrong side of a 7-1 defeat when they faced Barca in Spain in 2012, when Messi scored five, showed no fear as they went out to attack.
Javier Hernandez got in front of the Barca defence and forced a good save from Ter Stegen inside two minutes.
Ivan Rakitic headed straight at keeper Bernd Leno and then a Sandro strike was also saved, but Leverkusen continued to pose a threat.
A Kevin Kampl shot inside the area went just wide before the goal from Papadopoulos, who headed home from a couple of yards out with Ter Stegen not moving on his line.
A deflected Neymar shot came off the post and a Sandro follow-up was blocked on the line by Papadopoulos as Leverkusen hung on to their lead.
After the break the chances began to pile up for Barca, with Sandro and Neymar firing wide while Pique headed at the keeper when well placed.
To make matter worse for Barca they lost Andres Iniesta to injury in the second half.
Still, there was time for a thrilling finale for the Barca fans.
Sergi was quickest to the rebound after Leno's low save, and then Suarez came up with another telling contribution, sending a rising right-foot shot into the keeper's top corner.
(This story was refiled to add dateline)
(Reporting by Tim Hanlon, editing by Stephen Wood)
