MLB Rule 10.19 specifies in relevant part that in order to earn a save, the following must be true of a relief pitcher:
(a) He is the finishing pitcher in a game won by his team;
(b) […]
(c) He is credited with at least a third of an inning pitched; […]
Under what circumstances could the final pitcher of a game won by his team not be credited with one out?
Possibly if he’s called in and he successfully appeals a prior runner as not having touched a base? Is he then credited with that out?
Sudden rainstorm, blackout, etc.
what about a home team pitcher pinch hitting for the pitcher who pitched the top of the inning, then the winning run is scored while or after he pinch hits?
If he throws only the third strike to the batter? I think he still gets the out though, so I think I’m wrong.
Yes.
He would be the winning pitcher, and would not qualify for a save.
Yes, a pitcher finishing a strike-out gets credit for the strike-out.
This is the correct answer. It isn’t as uncommon as it might seem, since the delay attendant upon a pitching change provides umpires a chance to re-assess conditions and will sometimes lead to a game being delayed and eventually terminated.
Catcher picks off a runner trying to steal for the last out.
ETA under this scenario a closer could come in with two outs and a runner on first, throw one pitch high for ball one, have the catcher nail the runner trying to steal and the game would be over with the relief picher throwing only one pitch and not getting credited with an out (or even a strike).
I’m pretty sure the pitcher of record still gets credit for that.
I would be surprised if this is true. Do you have a cite?
Why wouldn’t he? If a catcher picks off a runner in any other circumstance, the pitcher gets credit for the out.
I don’t believe that’s true. Cite?
Man, are you a pain in the ass. 
Down about a tenth of the way is a note defining the number of inning pitched:
All it requires is a putout, and doesn’t state that the putout has to be done as a result of the pitcher’s actions. Putouts are further defined in section 10.10
Yes, but the whole issue is whether a pitcher gets credited for a putout if the out is made between pitches. I don’t think your cite gives a definitive answer to that.
For the record, I’m not trying to dig in my heels and say you’re wrong. You may very well be right. I really am just asking for a definitive answer to the question. I tried to search for one myself, but it’s kind of an esoteric thing to Google for.
It’s not something that’s going to be spelled out exactly in the rules of baseball, but in every game I’ve ever seen the box score for, the sum of the number of innings pitched for all the pitchers on one team equalled nine (or however many innings the game went).
For instance, take a look at the box score from the All Star Game this year. The game went 14 and 2/3 innings for the National League pitchers (the AL scored before their third out, but it was the bottom of an inning), and, sure enough, the total innings for all pitchers combined for the National League is 14.2. However, Milton Bradley got picked off at first, and the inning pitched was credited to Zambrano, despite the fact that the put-out was made between pitches.
Now that I look over the thread, DtC, I think I see the confusion (perhaps). The pitcher doesn’t get credited for a put-out, but still will get credit for pitching a third of an inning. He qualifies under 10.19.c as the saving pitcher, but doesn’t get credit for an out.
I think I understand the scenario you’re positing; correct me if I’m wrong: with two outs in the ninth, the winning team brings in a new pitcher with a runner on base. Before that new pitcher has a chance to throw a pitch, the catcher picks the runner off, ending the game. Is that right?
If so, I don’t think such a scenario is even possible. The ball is dead until the new pitcher is in, warmed up, and on the rubber ready to deliver. How is the catcher even getting the ball without the pitcher throwing it to him?
Dio had the pitcher throw one pitch (a ball, perhaps) before the runner gets picked off.
Maybe so. That’s not entirely clear to me, but it’s early. If catchers picking off runners didn’t count toward innings pitched, though, it would make scoring kind of a mess.
For any type of out recorded in any manner–pickoff, caught stealing, runners passing on the bases, interference, appeal plays, batter hitting a triple and getting thrown out at home–the pitcher in the game at the time gets credit for a third of an inning pitched.
Ok, I think I see what your saying. The pitcher doesn’t have to get credited with a put-out, he just has to be on the mound when it happens to get credited with a third of an inning. Makes sense.