Reading continued to dominate play, but fell behind in the 36th minute when Conway capitalised on poor defending from the hosts and guided the ball beyond Alex McCarthy.
Jordan Rhodes came close to doubling Blackburn's lead ahead of half time, but the Scotland international's header was well saved by McCarthy.
The match turned scrappy after the break, although the Royals almost drew themselves level in the 69th minute through substitute Royston Drenthe, whose side-footed effort from Jordan Obita's cross flew narrowly wide of the mark.
Drenthe came close again in the 73rd minute, lashing a strike over the bar from just outside the box, moments after Conway had missed an opportunity at the other end from a similar distance.
Blackburn defended in great numbers in the latter stages and frustrated their opponents, a tactic which paid off as the visitors held out through nine minutes of stoppage time to take maximum points.