Messi was absent due to a hamstring problem, whilst Suarez completed a two-game Cup ban, but the visitors still took control of the tie with two goals in six first-half minutes from Munir El Haddadi and Neymar.
However, Aritz Aduriz's 23rd goal of the season a minute from time handed Athletic hope for next week's second leg.
La Liga leaders Atletico Madrid also have work to do to reach the semi-finals after a 0-0 draw at Celta Vigo in their quarter-final, first leg.
“The (late) goal complicates the tie a bit for us, but I think we played a great game and now we have the return leg at home,” Neymar told Canal Plus.
“There is a bit of danger because they are a good team with players that can turn the game. We need to be concentrated.”
“We lacked fluidity in the second-half to do them damage,” said Barca boss Luis Enrique.
“Independent of the goal conceded, I thought the tie wasn't over at 2-0. I go home happy, but we have an interesting game next Wednesday.”
Enrique's men had hammered a 10-man Athletic 6-0 on Sunday as well as 3-1 in last year's final, and there was a familiar feeling for the Basques despite a bright start from the home side.
The European champions went ahead on 18 minutes thanks to a brilliant team goal that started with goalkeeper Marc-Andre ter Stegen and ended with Ivan Rakitic's inviting low cross being swept home by El Haddadi for his fifth goal in the competition this season.
Athletic contributed to their own downfall moments later, though, as goalkeeper Iago Herrerin missed the ball as he came flying out his goal and allowed Neymar to roll his 20th goal of the season into an empty net.
The hosts dominated the second period but looked set to end the night empty-handed when Ter Stegen made a brilliant save to turn Aduriz's header over 10 minutes from time.
However, Athletic finally got the goal their display deserved when Aduriz turned home Mikel San Jose's pass.
And Ter Stegen had to make another fine stop from Aduriz seconds later to prevent Athletic salvaging a draw in stoppage time.
“They had two chances and scored two goals, that is why they are so good,” said Aduriz.
“We had a lot of dangerous situations, but we couldn't put them away.”
Celta had the better of a game of few clear-cut chances at Balaidos, but Miguel Angel Moya saved Iago Aspas's effort before Atletico captain Gabi cleared off the line late on.
“The tie is open, but a goalless draw is better for them,” said Atletico coach Diego Simeone.
The sides will meet for a third time in 17 days next Wednesday at the Vicente Calderon for a place in the last four.
Gary Neville's Valencia host Las Palmas and Mirandes, the only second-tier side left in the competition, visit Sevilla in the other two quarter-finals on Thursday.– AFP
Original source: Barca make do without Messi, Suarez