The Spanish champion dominated the game, but the Italian team played its part by fighting back from a goal down and seeking to match the complex spells woven by Barcelona's talented players. But Juventus could not hang in there when Barca accelerated into an unmatchable rhythm.
Watched by more than 70,000 people in Berlin's Olympic Stadium, Juventus striker Alvaro Morata cancelled out Ivan Rakitic's early opener, Messi set up the vital second for Suarez, and Neymar secured Barcelona's fifth European Cup deep into stoppage time with the last kick of the game.
The Messi-Suarez-Neymar axis complete the season with a combined 122 goals — and now European winners' medals to join their league and cup ones.
Three years after Pep Guardiola left Barcelona and after last season ended without a trophy, Luis Enrique matched Guardiola's 2008-09 feat of winning a treble in his first season in charge, suggesting the Catalan giants could be ready for a new period of European dominance.
"A magnificent, spectacular day," Enrique said.
With a first Champions League triumph since 2011, Barca has won European soccer's top trophy four times in the last decade.
Andres Iniesta has featured in each final and the midfielder is also celebrating his second treble. Iniesta played a deep defensive midfield role in Berlin as the strikers caused havoc in the panicky Juventus defence.
"We won't see this happen again very often," Iniesta said. "Six years later we did (the treble) again. All my words fall short. Let the people enjoy this Barca."
Few will be enjoying the victory more than Suarez.
The Uruguay striker began the season in shame, signed from Liverpool for $110 million despite being handed a four-month ban by FIFA for biting an opponent at the World Cup. He powered in Barcelona's second goal in the 68th minute after Messi's shot was palmed away by goalkeeper Gianluigi Buffon.
"When you come to a team like Barca, you know that you have come to win," Suarez said. "This is a dream."
Suarez is rarely far from controversy, however. Twice he writhed around in agony, winning free-kicks and feigning serious injury. Juventus fans yelled derision when he recovered miraculously after wasting minutes.
Just like Suarez, Neymar has collected the top prize in club soccer for the first time after two years at Barcelona.
Neymar had a goal disallowed for handball but, with the final going into a seventh minute of stoppage time, the Brazil striker found space to score again as Juventus desperately sought a leveler, leaving its defence bare. This time it counted.
For Juventus there was no third European title, just the unwelcome record of being the first team to lose six finals. Still, the season ends with a domestic double
"While we've played a lot of finals, a lot of us had never played a Champions final," Juventus midfielder Claudio Marchisio said. "There was a bit of fear."
The Italian champion initially looked like it could cause an upset against the most dominant team in Europe in the modern era. Assertive from kick off, Juventus made Barcelona look anxious initially. It didn't last long — just four minutes, in fact.
In a delightful move that involved all 10 outfield players touching the ball, starting with Jordi Alba's throw in, Barcelona took the lead.
So often the provider of goals, Messi delivered a precise cross-field pass in the goalward advance. It landed at the feet of Jordi Alba, who played in Neymar and the ball was then slipped through to Iniesta.
With a flick, Iniesta squared for Rakitic to score from close range past Buffon.
Had it not been for Buffon's stretching save at the start of the second half to deny Suarez, Juventus would have been two behind. Messi, for once, also couldn't find the target after a one-two with Suarez.
It proved costly. A back-heel from Claudio Marchisio began the move that led to Juve's equalizer.
An initial shot from Carlos Tevez was parried by Marc-Andre ter Stegen. But Morata, who scored in both legs of the semifinal against former club Real Madrid, sent the follow-up into the net with ease in the 55th minute.
Juve didn't use the opportunity to swing the final in its favour, and Barca was back in front 13 minutes later after launching an electrifying counterattack.
Within five minutes of Suarez's goal, Barca was celebrating again — but the joy was short-lived. Neymar was spotted to have headed the ball onto his hand before it went past Buffon.
But Neymar did have the final say in this final with the late, late goal.
It gave Xavi Hernandez the perfect send-off, departing for Qatar after spending his entire career at Barcelona with another medal.
"Nothing can beat finishing in this way," the 35-year-old midfielder said.
Now the soccer world's focus is likely to shift back to allegations of corruption, arrests and investigations that have pummeled the game's reputation in the last 10 days.
Rob Harris can be followed at www.twitter.com/RobHarris