Messi had the ball in the net on the half hour, but the goal was disallowed for an earlier infringement, and Juve finished the opening half with a flurry.
Miralem Pjanic fired Cudrado’s cutback high and wide, while Gonzalo Higuain and Leonardo Bonucci both had Ter Stegen scrambling to save in the final minutes of the half.
Luis Enrique took off French left-back Jeremy Mathieu and brought on Portuguese midfielder Andre Gomes at half-time, moving Javier Mascherano back into defence.
The visitors briefly showed promise, with Suarez firing just wide, but Juve went further ahead on 55 minutes.
Chiellini, famously bitten by Suarez when Italy played Uruguay at the 2014 World Cup, held off Mascherano at a corner to send an angled header in off the far post.
It was his second goal in the competition and first since October 2009 when his header secured the points against Maccabi Haifa in Turin.
Suarez fired just wide again midway through the second half and Chiellini was lucky not to concede a penalty when both he and Cuadrado dived low to thwart a shot which came off the defender’s face then rolled up his arm.
Barca continued to press but could not get the away goal that would have improved their chances ahead of the second leg.