As others have stated, you have to go well back in time to find the roots of the Dead Puck Era. I think we all know the 1979-1986 (post-WHA; 18-year-olds drafted and playing) era was characterized by a disproportionate number of young players, as well as sudden developments in offense and skating, leading to higher scoring levels from about 1981 to 1986. NHL offense (Gretzky-era) had taken a lot from Soviet hockey and European style of the 1970s, while goaltending and defense was weaker due to so many young players on the blue-line and in goal.
Then, things started happening that led to the Dead Puck Era:
1986 playoffs: Montreal and Patrick Roy win the Cup, with a defense-first team that rolls four lines. This was the first Cup since 1974 that wasn't a repeat winner or a Dynasty team.
1986-87 season: Scoring suddenly drops across the board. It's not often noted, but this season had the most parity since the early 1940s (or something). (Scoring would bounce back again the next two seasons --- though not back to 1981 to 1986 levels --- but then from 1989-90 onward would fall.)
1988-89, 1989-90, 1990-91 seasons: The clubs that were 1st and 2nd in defense (1989), 1st in defence (1990), and 1st in defence (1991) finished at the top of the standings.
So, now, developments in goaltending and team defensive strategy are shown to allow medium-skilled clubs to succeed against higher talented clubs (which rarely happened in the 70s or early/mid-80s).
At this same moment, huge NHL salaries---without a salary cap---start to become a thing. This leads to two more important aspects: (1) Some clubs can afford big salaries (Rangers) and some can't (Edmonton, Winnipeg, Quebec). If you can't afford big salaries, it's now been shown that medium-skilled clubs can succeed against higher talented clubs with strategic defence and good goaltending. And (2) Veteran star players have their careers extended by virtue of singing lucrative long-term contracts. This means that the 1970s/80s norm of star players retiring, or being considered washed-up and sent to the minors (Marcel Dionne) by age 33 or whatever, is over. Now, NHL players will have longer, more lucrative careers, often playing into their late 30s and even 40s. And older players on with big roles on teams means more defense.
1995: New Jersey wins Cup.
1997 & 1998: Detroit wins Cups with several older players and lots of defense.