I think the most likely interpretation of the phrase is that when a situation "has come to the triarii," it has become dire. The situation has breached two safety nets and is now hangs upon the last "line of defense." The hastati and then the principes have both failed to do their jobs, and we get the impression that only rarely did the triarii have to get involved and set matters straight. Livy (8.8) tells us that the hastati were the young-uns of the army, followed in the second rank by the principes (and these names make no sense in this arrangement, I know). The triarii were the old-timers, the most experienced, and typically rested on one knee during the battle (also 8.8). Actually, now that I look at it, 8.8 is the source of the phrase as well!

There is a reference in the comic poet Plautus to the triarii which suggests they were well-known for hardly ever exerting effort:

Agite nunc, subsidite omnes quasi solent triarii! ("Come now, everybody sit down just like the triarii are always doing!")(Varro L.L. 5.89)

As for examples of battles involving the triarii, Livy conveniently provides an example of the triarii turning the tide of battle just a little bit below the source of the phrase itself, during a description of a fight between the Romans and the Latins (Book 8 deals with the First Samnite War and the Settlement of Latium, 341-321 BCE):

When Manlius heard the fate of his colleague, he honoured his glorious death with tears no less than with the due meed of praise. Meantime the battle proceeded, and in some quarters the weight of numbers was giving the advantage to the Latins. For some time Manlius was in doubt whether the moment had not come for calling up the triarii, but judging it better for them to be kept fresh till the final crisis of the battle, he gave orders for the accensi at the extreme rear to advance to the front. When they came up, the Latins, taking them for the opposing triarii, instantly called up their own. In the desperate struggle they had tired themselves out and broken or blunted their spears, but as they were still driving the enemy back by main force, they imagined that the battle was decided and that they had reached their last line. Then it was that the consul said to his triarii: "Rise up now, fresh and vigorous against a wearied foe; think of your country and your parents and wives and children; think of your consul lying there dead that ye might win the victory!" They rose up fresh and resplendent in their armour, as though a new army had suddenly sprung up, and after letting the antepilani retire through them they raised their battle-shout. The front ranks of the Latins were thrown into disorder, the Romans thrust their spears into their faces, and in this way killed the main support of their army. They went on without being touched through the remaining companies as though through a crowd of unarmed men, and they marked their advance with such a slaughter that they left hardly a fourth part of the enemy. The Samnites, too, who were drawn up close to the lowest spurs of the mountain, were threatening the Latins on their flank, and so adding to their demoralisation. . .

[but the Latin leaders said that] the Romans enjoyed no more than the name of victory, in every other respect they were as good as defeated. The headquarters of both consuls were polluted with blood; the one had murdered his son, the other had devoted himself to death; their whole army was massacred, their hastati and principes killed; the companies both in front of and behind the standards had suffered enormous losses; the triarii in the end saved the situation. The Latin troops, it was true, were equally cut up, but Latium and the Volsci could supply reinforcements more quickly than Rome. . . [Livy 8.10-11