Well, "saved" is a vague word and, let's face it, it did very little militarily to help Prussia (with an excuse that the whole affair was over too soon). Not sure where exactly did you get your data about the territorial swaps.
Defeat at
Königgrätz was humiliating but it was not Jena-like catastrophe. Prussians still had their 2nd Army coming and, with a brilliant maneuver executed by von Blumenthal, it managed to stop Austrian advance and guaranteed safe retreat to Saxony with the part of the 1st Army getting back to the right bank of Oder River in Silesia (and having a safe line of retreat to Posen). Austrians simply did not have enough troops to both take Silesia (their initial goal) and safely contain Prussians in Saxony on their left flank. The war started turning into a stalemate thus giving other two continental Great Powers time to interfere and, as you put it, "save" Prussia (theoretically, Prussia should win a protracted war thanks to its greater economic resources but nobody was going to test this theory).
In the intensive diplomatic exchange France and Russia (none of which had a single reason to have any good feelings toward Austria) demanded from Austria to agree to a peace with the minimal territorial concessions from Prussia. As a demonstration of their seriousness France assembled 200,000 troops on her northern border while Russia moved 120,000 to its border with Austria (Galicia) and 50,000 to Kalish to back up Prussian troops in Silesia. In a meantime Prussia continued mobilization of its reserves. Austria had to start talks.
By the Peace of Paris all combatants more or less returned to their pre-war borders: Austria got Neurode - Glatz - Habelschweirdt region, Prussia - Leipzig area and France got Southern part of the Rhineland (not sure where did you get these greater swaps). Alexander II got a moral satisfaction by repaying Austria for its "ingratitude" toward his father.
Immediate results:
1. Bismark and Moltke had been sacked. Von Blumenthal had been promoted to the Chief of the General Staff. Roon, somewhat discredited, survived and military reforms continued.
2. Agreement regarding Franco-Russian-Prussian military alliance had been signed. A secret protocol stated dismemberment of the Austrian empire as a goal for the possible future war.
3. Austrian militaristic faction led by Austrian Chancellor
Count Friedrich Ferdinand von Beust was openly unhappy with the results of Peace of Paris and called to a new decisive war with Prussia and made threatening noises against those "who stabbed Austria in the back" (France and Austria). In euphoria of victory nobody thought about rearmament or military reform.
In 1870 Prussia used an artificially created diplomatic scandal to provoke Austria into a military action. Contrary to the rosy expectations of the "militarists", Austria found itself confronted by the alliance of Prussia-France-Russia (so-called "Tripple Alliance") with Piedmont joining as soon as the war started. Austrian German allies, Bavaria, Hanover, Saxony, Wurttemberg, etc. proved to be of a little use.
Resulting so-called "6 weeks war" ended up with a complete defeat of Austria by the Tripple Alliance:
1. France got the left bank of the Rhine.
2. The German Empire had been created (Tripple Alliance had been renamed into "Alliance of Three Emperors") with King of Prussia assuming the imperial title. Kingdom of Prussia got back Silesian territories lost in a previous war.
3. Piedmont united all Italian lands to became Kingdom of Italy.
4. Russia got Galicia (and had to figure out what to do with the local Jewish landowners).
5. Bohemia and Hungary (Francis Joseph I did not bother with being crowned in either) had been proclaimed the independent kingdoms (suitable German princelings related to both Hohenzollern and Holstein-Gottorp family got the crowns).
6. Not to be forgotten, Britain expressed its unhappiness.