The visitors could have taken the lead with just two minutes on the clock, but Mario Mandzukic fluffed his lines from close range.
Croatia continued to do most of the pressing during the remainder of the first half, though they found it difficult to carve out any clear-cut chances.
The chances fell for both sides after that, with Subasic thwarting Ruslan Gurbanov and Mandzukic missing a header at the other end of the pitch.
Croatia started to take control during the final stages as Mandzukic's point-blank effort was saved by Agayev before Mateo Kovacic's shot flew the wrong side of the post.
The away side had chances to win the match late on, but Marko Pjaca fired wide after Mandzukic had squandered yet another opening inside the box.