The Uruguayan can rarely have scored more important goals for the Catalan giant, which trailed to a 25th minute goal from Fernando Torres, who then turned from hero to villain by promptly getting himself sent off for two rash challenges.
Barcelona, so lackluster in the first half and with its defense of the Champions League under real threat, finally woke up after the break, but it took until the 63rd minute to draw level.
Neymar hit the woodwork and wasted two other clear opportunities before Suarez, quiet until that point, scuffed home his effort after Jordi Alba's mishit shot found him in front of goal.
With Atletico now pinned back, Barca's second was almost inevitable and 11 minutes later it was Suarez who met a Dani Alves volleyed cross with a bullet header.
Atletico held out until the end, content with the away goal ahead of the return in its Vicente Calderon Stadium in eight days time.
It proved a frustrating night for Barcelona's talisman Lionel Messi, seeking his 500th goal for the club, a clever overhead effort in the second half the closest he came to scoring.
But it could so easily have been worse for the home side, still reeling from its El Clasico defeat to 10-man Real Madrid Saturday.
Atletico, so resilient in defense, switched quickly to attack and Koke played in Torres to score his 11th goal against Barcelona.
The visitor continued to look dangerous on the break but all the good work was undermined in the space of six minutes as Torres impeded Neymar and then received a second yellow card for bundling into Sergio Busquets.
His coach Diego Simeone, taking charge of Atleti for the 250th time, was incensed but the moment of madness was to cost it dear.
Neymar's efforts, with prompting from Messi, softened Atletico up before Suarez struck for his 44th and 45th goals of the season and eighth in the Champions League in eight games.
Barca boss Luis Enrique said his side would not sit back on its lead in the second leg.
"We will go to the Calderon to win. If there is a team that can do it, then it is us," he told the official Barcelona Twitter feed.
Bayern Munich will also have a slender lead going into the second leg of its tie with Benfica after a 1-0 win in the Allianz Arena.
Arturo Vidal scored after only two minutes with a clever header from a Juan Bernat's cross and further goals were expected.
But the Portuguese visitor defended strongly and might have equalized in the second half but for Javi Martinez blocking a Jonas effort,
Late on, Bayern's top scorer Robert Lewandowski burst clear but his pass to an unmarked Philipp Lahm was overhit, summing up his side's night.
"Benfica's back four stood up well and we missed out on that second goal," Bayern's goalkeeper Manuel Neuer told Germany's Sky TV.