It was first introduced in Tomb Raider Chronicles where Lara was able to balance on Tightropes. This, however, was achieved with the use of a special object that triggers Lara's balance animation. To use the tightrope, the player had to press the Action button.
Balancing was reintroduced in Tomb Raider: Underworld, where it could be used to climb higher up. It was kept on as a move in the later games, the 2013 reboot and its sequel Rise of the Tomb Raider. Here Lara can balance on branches, beams, and other narrow objects.