McLeish is delighted to sign the former Arsenal man and thinks he has added a "top quality player" to his squad".
"Hleb is a top quality player - one who at one stage we'd only actually dreamed about really," he told the club's official website.
"Over the past year or so we have talked him but with no real belief we could get him. We decided to give it a try and towards the end of the window we found it might be possible.
"This is a major signing for us and we are very happy to have him here. He is an extremely creative type, who is skilful, a great dribbler and has an excellent touch and quick feet.
"As with all the players we have brought to the club in recent times I'd like to thank the board for their support in making this happen."