The home side applied early pressure as they looked to set up a goal-scoring opportunity in the opening minutes of the game, but David Marshall was able to claim several crosses before the hosts could take a shot.
The league leaders had had few chances early in the first half, but a counter-attack in the 27th minute resulted in Kim Bo-Kyung feeding the ball through to Conway on the edge of the area, with the Scot smashing it past Lee Grant.
The goal lifted Cardiff, and they threatened to double their lead before the break, with Burnley's loan striker Sam Vokes having to drop back to help in defence.
Conway looked for his second in the final minutes of the half, taking another shot from the edge of the box, but this time he could not keep the ball down.
If the score stays the same in the second half, Cardiff will be crowned Championship champions today.