Queen beats rook, but not by chasing it. Centralise the king, drive the enemy king to the edge, force the rook next to its king, and reach the zugzwang positions where the rook must move away and be lost.
Queen versus rook is a theoretical win, yet many strong players have failed to convert it in practice. The reason is that the rook is hard to catch while it has room and the attacker's king is far away.