Question about triggered abilities and missing the timing

I cast a spell that would activate a triggered ability. When the triggered ability activates, it goes on the stack. An opponent attempts to counter the spell.

Wouldn't the counter miss the timing for my spell, since there's a triggered ability in the way? Isn't this how spell after spell after spell works? I can't seem to find anything about this under 405 or 603, so how exactly does this all work?