After watching some "Batman: The Animated Series" DVDs, I hunted around the internet and found this article on the music for "Batman" and the other WB animated superhero shows: "Knowing the Score."
One of the biggest factors, perhaps the biggest factor in the excellence of the '90s Warner Brothers TV cartoons was the use of a full orchestra and new, original scores for every episode. This was previously unheard-of in TV cartoons, which almost always used stock library music, or a catalogue of music cues that were recycled from episode to episode, or electronically-generated music. But when WB did "Tiny Toon Adventures" in 1990, Steven Spielberg and supervising composer Bruce Broughton wanted to get a Carl Stalling sound for the show, so the decision was made to create new scores for every episode, give a composer two weeks to write a score for a twenty-minute episode (not a lot of time to write 20 minutes of music, but more time than TV composers generally get), and record the music with an orchestra of about 27 players. Don Davis, one of the composers on the show, described the scoring process in this interview.
Fortunately, when WB followed up "Tiny Toons" by putting "Batman: The Animated Series" into production, they continued to use the same full-orchestra format, though an action-adventure show required a different kind of score: not every scene had music (whereas Stalling-style scores hit every beat of every scene), and the music wasn't supposed to sound like cartoony music. The choice for supervising composer of "Batman" was Shirley Walker, who was the orchestra conductor on the Batman movie; other regular contributors included Michael McCuiston, Harvey Cohen and Lolita Ritmanis. Just as the "Batman" cartoon series was more satisfying overall than Tim Burton's Batman, Walker's scores were arguably better than Danny Elfman's score for that movie (which is where the show's theme song came from). She came up with timeless-sounding, near-symphonic scores that fit in perfectly with the timeless look of the show, and created memorable leitmotifs for most of the characters; I particularly liked the demented Mad Hatter theme and the Harley Quinn theme, which could either be wacky and funny or sultry and romantic depending on the mood of the scene.
The most prolific composers on "Tiny Toons" included William Ross (now a successful feature composer) and Richard Stone, who became WB's first choice for all comedy cartoons; he was the supervising composer of "Animaniacs," "Pinky and the Brain" and "Freakazoid!" among others. Stone had an intimate knowledge of Carl Stalling's work and style, but he brought his own innovations and quirks to the music, as did the other composers in his stable: Steve Bernstein, Julie Bernstein (a husband-and-wife team) and Carl Johnson. Writing scores for these TV comedy cartoons was in many ways an even harder job than scoring theatrical cartoons. Whereas the classic WB cartoons were timed to a pre-determined beat, allowing Stalling to establish a consistent meter for every scene, the TV cartoons were more haphazardly put together, meaning that they could alternate one shot in one meter with another shot in a totally different meter. This made it tougher for the composers to write music that could mimic the action and still be recognizably musical -- yet they managed to do it, and do it fantastically well.
Sadly, Stone died from pancreatic cancer in 2001. this article had some more information about him and his approach to scoring cartoons.
As is mentioned in the first article I linked to, around 2000 the budget of WB's cartoons was cut, meaning that they could no longer use full orchestral scores. "Justice League" does the best it can with computer-generated scores, but there's just no substitute for a real orchestra, and the lack of acoustic music does give the newer cartoons a less classy, less timeless feel than the '90s stuff. That's why the '90s cartoon scores, and the composers led by Walker and Stone, are so special: for one decade, a TV cartoon studio was turning out cartoons that sounded as classy as great feature cartoons.