Kyriakos Papadopoulos had headed the visitors into a deserved half-time lead as Barca struggled to open up the Germans without Messi's magic touch.
However, despite also losing Andres Iniesta to a hamstring injury, Suarez turned the game around in the final 10 minutes as Sergi Roberto tapped home the rebound after the Uruguayan's initial effort had been saved.
And the former Liverpool striker thumped home a stunning effort to hand Barca a hard-fought three points, two minutes later.
Victory takes Barca top of Group E, a point ahead of Leverkusen and BATE Borisov, the latter being 3-2 winners over Roma on Tuesday.
"We believed in ourselves until the end," Suarez told beIN Sports.
"We couldn't play great football, but we had to win the game. We had to show we are the champions."
Messi was in attendance at the Camp Nou, but his teammates made a shaky start without him on the field as former Real Madrid striker Javier Hernandez forced Marc-Andre ter Stegen into a smart save at his near post inside two minutes.
Ivan Rakitic had Barca's first effort on target, but the Croatian placed his header too close to Bernd Leno.
Sandro Ramirez was the man given the unenviable task of replacing Messi and nearly repaid coach Luis Enrique's faith as he played a lovely one-two with Suarez before Leno pulled off a spectacular save to turn the 20-year-old's effort behind.
However, Leverkusen's speed on the break was a constant threat and Kevin Kampl should have done better on two occasions as he curled the ball wide and was then denied by a brave block from Gerard Pique with an even clearer sight of goal.
Leverkusen finally took a deserved lead on 22 minutes when Hakan Calhanoglu's brilliant in-swinging corner caught Ter Stegen off guard and Papadopoulos nodded home.
The German international made amends moments later as he stood up to prevent Karim Bellarabi from doubling the visitors' lead.
Much had been expected of Neymar in Messi's absence, but the Brazilian had been kept very quiet until his deflected effort hit the post six minutes before the break and Sandro's follow-up effort somehow deflected off Papadopoulos and behind for a corner.
Hernandez had a golden chance to put the game beyond the hosts at the start of the second-half but blazed over when unmarked just 12 yards out.
At the other end, the chances began to arrive for Barca as Sandro again fired wide before Neymar pushed an effort off target.
However, the bad news continued for Enrique when Iniesta pulled up just after the hour mark to add to the Catalans' mounting injury crisis.
Enrique was forced to turn to youngsters Munir El Haddadi and Roberto from the bench to turn things around.
Both made a big impact as Barca reacted like champions in the final 10 minutes.
Roberto pounced after Leno had saved Suarez's initial effort to net his first European goal.
And moments later substitute Munir squared for Suarez to smash the ball high past the helpless Leno for a memorable winner.
By Kieran Canning