The visitors were the first to threaten when Sergey Ignashevich brought the ball forward from the back and was allowed to raid to the edge of the box, but he sent his effort over the bar.
Aleksandr Kokorin should have given Russia the lead in the 28th minute when he latched on to a cross from the right, but he failed to guide his header from close range into the back of the net.
In the second half, Russia continued to threaten the Moldova goal as Pavel Mamaev latched on to a loose ball on the edge of the box, but his right-footed drive sailed wide of the post.
Kombarov nearly doubled his side's advantage minutes later when he was allowed to shoot from distance, but Koselev made a solid stop to deny him.
Russia secured their win in the 79th minute when Ignashevich guided the ball into the path of Artem Dzyuba at the back post, who slotted home from one yard.
Eugeniu Cebotaru got a goal back for Moldova late in the game, but the visitors held out for the three points to edge closer towards a place in Euro 2016.