The Real Ghostbusters/Heartwarming

The ending of "The Thing in Mrs. Faversham's Attic": proof that Peter Venkman really does have a heart. And, well, the entire climax of "Janine, You've Changed" for Egon/Janine fans. Like the Awesome Moments, the show didn't have very many of these (since the movie didn't really focus on them), but when they did appear they were even more poignant.
The ending of "Look Homeward, Ray." Ray is his hometown hero again, he's back with the Ghostbusters after having quit earlier in the episode, and the girl he always had a crush on as a kid kisses him.
The end of "Knock Knock." After Peter yells at Slimer for pigging out (again), the others suggest cutting him some slack - with Egon pointing out he's a ghost living with a bunch of Ghostbusters. That night, Peter sneaks Slimer a mini-pizza.

"You breathe a word of this to anyone, and I'll deny it."

In "The Brooklyn Triangle," the Collector's forces steal objects throughout New York, which leads to Winston and his father being captured. While in his fortress, they have the time to hash out their falling out over Winston leaving the construction business. They patch things up just in time for the Collector to say his sole motivation was finding the ancient key that would free him from the fortess.

Winston: What do you think, Dad?
Ed: I know when somebody's lying, and I believe him.
Winston: Yeah, so do I.
Ed: Then that's good enough for me. My boy oughta know. He's a Ghostbuster.

In "Slimer, Is That You?", well-after the Body Swap has taken place:

Janine: It's nothing, Egon. Just realizing that it really is your mind I like.

At the end of "Troll Bridge," Peter tricks the trolls into thinking they've turned the runaway Wart into a statue with the particle beams - ending their threat against New York and allowing the little guy to do whatever he wants to from now on. As the guys see Wart off, he says his first real English phrase: "Thank you, Peter."
The ending of "Who's Afraid of the Big Bad Ghost?" when Uncle Horace remembers what he was looking for and says his final goodbye to Olivia before dispersing peacefully.