It appeared as though the visitors would take a slender lead from the quarter-final encounter at Camp Nou on Wednesday after Atletico's Brazilian midfielder struck.
Diego had started on the bench, but was called upon in the first half when Diego Costa – who overcame a knee injury to start the match – had to exit play.
However, Brazil international Neymar ensured Gerardo Martino's side would not head into the return leg at Vicente Calderon with a deficit when he curled past the otherwise excellent Thibaut Courtois with just under 20 minutes remaining.
Diego Simeone's men will still have the advantage of an away goal in that fixture, but the tie remains nicely poised after the first encounter.
Atleti were hit with an injury blow of their own as Costa limped off and a subsequent lack of physical presence in their attack helped Barcelona dictate the tempo, but Messi’s tame five-yard header was the closest they went before the break.
Indeed, Martino's men were almost caught on the counter, Villa's curling effort from the edge of the box forcing Pinto into a smart low save on the stroke of half-time.
Barcelona sought a quick response, and Sergio Busquets' well-controlled volley from 20 yards almost provided one shortly after the hour mark, but Courtois was equal to the task.
Iniesta's exquisite outside-foot throughball found the 22-year-old on the left-hand side of the area and he curled a fine strike past a despairing Courtois.
The equaliser buoyed Barcelona, and the visitors had Courtois to thank for ensuring a fourth draw of the season between the two sides, the Chelsea loanee making impressive saves from Iniesta and Messi, the latter of which was a fine diving stop to prevent the Argentina international's superb free-kick finding the net.
