It is now the sixth straight season that the Gunners have been eliminated at the round of 16 stage after their latest 5-1 aggregate defeat.
Following Arsenal’s promising start in the Catalan capital, Neymar put the hosts ahead after the north Londoners gave possession away cheaply.
January signing Mohamed Elneny then drew Arsenal level early in the second half when he rifled a superb 18-yard effort into the top corner.
With two minutes left on the clock, Lionel Messi showed his classy touch to flick the ball with the outside of his boot past Ospina, as the Blaugrana advance to the last eight.
As Wenger’s men entered the game trailing 2-0 on aggregate, his selection raised eyebrows when he named his starting XI – dropping Per Mertesacker and handing starts to Elneny, Alex Iwobi and Mathieu Flamini.
The north Londoners started brightly, with Mesut Ozil and Elneny going close as they sent 20-yard strikes across goal and past the post.
David Ospina produced an outstanding save to deny Messi from point-blank range as the Blaugrana grew into the contest.
The promising start was all for nothing as Laurent Koscielny was punished for playing a poor ball out of defence. The home side regained possession and made the Frenchman pay as Suarez slid the ball through to Brazilian forward Neymar, who neatly dispatched.
Arsenal’s best chance of the first half fell to former Barca playmaker Alexis Sanchez, who headed wide from six yards.
The Gunners’ travelling fans were able to celebrate early in the second half as Elneny made it 1-1 after 51 minutes.
The Egyptian netted his first goal for the club after meeting Sanchez’s pass on the edge of the box, striking past German goalkeeper Marc-Andre ter Stegen.
Welbeck could have doubled their tally, but for Javier Mascherano’s last-ditch challenge.
Barca were back in front as Dani Alves’ cross from the right was met by Suarez, who shaped his body before volleying first time into the top corner.
Welbeck hit the bar minutes after Suarez’s goal before ter Stegen made a superb double save to prevent Sanchez’s free-kick, reacting well to deny Olivier Giroud from the follow up.
Messi soon made it five on aggregate as he controlled the ball in the box, flicking the ball with the outside of his boot into the far side of the goal.
Elsewhere in the Champions League, talkSPORT2’s live commentary game saw Pep Guardiola’s Bayern Munich complete an impressive comeback over 120 minutes to reach the quarter-finals at the expense of Juventus.
The Serie A leaders – and last year’s beaten finalists – thought they had done enough to eliminate Bayern in their own Allianz Arena thanks to first-half goals from Paul Pogba and Juan Cuadrado.