What is a situation where saving just a small amount resulted in saving the mission, or some other activity or effort?

At the tactical level, attention to the small details will save lives.
The ambush of the 507th Maintenance Company in 2003 has become a schoolhouse narrative for that.
Attention to details like the primary route, reading a map, using commo and GPS would have prevented the break in contact and erroneous turns that got them into the ambush, and attention to simple weapons maintenance would have prevented their rifles from jamming at the start of the firefight. Those 5 small details would have saved 11 lives and prevented the capture of 6 more.

At the operational level, here's two examples.
Rental cars for schools, at least in the reserve compos but the budget concept is similar to active: Every time you send a Soldier to school, and pay for a rental car that wasn't needed (billeting, dining and classroom all in walking distance), what's the big deal? A rental car is convenient, and it's just $300-$3000 per school?
Each one eats up the cost of flights for sending 1-5 more Soldiers to schools. Extrapolate that for just one large state Guard, that could mean the difference of a few hundred more schools per year, across just the Army Guard, a few thousand schools, because someone didn't want to walk a half-mile.
Training ammo, again, this is a reserve compo example, but active is similar.
Every time your forecast and request ammo for training, that ammo is shipped to the ASP you need it at. That cost of shipping is routinely more than the actual cost of procurement. If you cancel that training, that ammo often has to be shipped again to another ASP where it will get used. Which means we have now spent 2-4 times the original cost of procurement on that ammo, due to inability to plan and manage training.

At the strategic level, across the DoD, the "why bother" mindset is how we waste as much as $125 billion of our budget every year. And that waste every year further delays procurement of needed equipment and delays or cancels needed training.

Not sure I am deciphering the question correctly but here goes. Attention to detail is effective in every aspect of the military. The easiest example for me is training and combat. We train gard and repetitively. NCOs and Officers ensure that we stope events to AAR ensuring that when the scenario is run again that it is done correctly, this goes on until combat. At this point the repetitive training coupled with Attention to Detail allows the organization to operate efficiently and safely with little oversight.

Lt Col Timothy Cassidy-Curtis During my civilian career, I would hold brown bag lunches on money management. I would help participants with topics that included budgeting, saving, living beneath their means, investing, etc. I would show how saving enough to receive company matching would make a huge difference. Above all, I strived to show attendees that managing money was not difficult and the rewards were worth the effort.