There are two major moments in The Kite Runner by Khaled Hosseini in which Amir knowingly and willingly allows Hassan to become a scapegoat for him, though Amir does the same thing to Hassan in small ways all the time.
The first of these major incidents is when Hassan is acting as Amir's kite runner in the local competition. Hassan has a gift for finding the winning kite, and Amir knows it. He also knows that Hassan will do whatever he must to gift Amir with the winning kite and that Baba (Amir's father) will be finally be pleased with Amir if he wins the competition.
So, when Amir sees Hassan being raped by Assef, a racist bully, he makes the deliberate choice to sacrifice Hassan in order to win his father's approval. We are certain Amir knows what he has done when he says this:
I actually aspired to cowardice, because the alternative, the real reason I was running, was that Assef was right: Nothing was free in this world. Maybe Hassan was the price I had to pay, the lamb I had to slay, to win Baba.
While Amir might not have been able to save Hassan from Assef, at least Hassan would not have suffered alone in his shame. The second time Amir sacrifices Hassan is more deliberate.
After the incident in the alley, Amir finds it difficult to live with his guilt and shame; Hassan's presence makes Amir uncomfortable and he tries to get Baba to send Hassan and his father away. When that does not work, he plants some things under Hassan's mattress and then accuses Hassan of stealing them. When Baba asks Hassan directly if he stole these items, Hassan does not waver; he falsely admits to stealing them, knowing that the consequences. Baba has no choice but to send Hassan and Ali away because everyone knows Hassan never lies. Not only did Amir get Hassan banished, he forced Hassan to go against his own moral code. For Hassan, though, Amir's welfare always came before his own.
This explains the first half of your discussion point, that Amir sacrificed Hassan; the second half of the equation is that, in doing these awful things, Amir also sacrificed something. By allowing these things to happen to Hassan, he diminishes himself. Because he does still have a conscience, he suffers the guilt and shame of his own cowardice. When he looks in the mirror, he does not like what he sees and knows that Hassan is a much better person than he is, despite the privileges of his wealth and ethnic status.
This is why the story begins with these lines:
I became what I am today at the age of twelve, on a frigid overcast day in the winter of 1975. I remember the precise moment, crouching behind a crumbling mud wall, peeking into the alley near the frozen creek. That was a long time ago, but it’s wrong what they say about the past, I’ve learned, about how you can bury it. Because the past claws its way out. Looking back now, I realize I have been peeking into that deserted alley for the last twenty-six years.”
The adult Amir is suffering the effects of his cruelty and cowardice so long ago. His past actions have shaped what he has become--and he does not like what he has become.
“It may be unfair, but what happens in a few days, sometimes even a single day, can change the course of a whole lifetime,” and this is true for Amir. He sacrificed himself--and Hassan--for a few moments of praise from his father and some relief from Hassan's presence. The adult Amir knows he made a bad trade because he also sold his own soul.