Ajax, winners of the UEFA Cup in 1992, held a 4-1 lead from the first leg in Amsterdam, and Dolberg's sixth Europa League goal of the campaign - a subtle dink over Anthony Lopes - furthered their advantage 27 minutes into a thrilling contest.
However, naive defending from Peter Bosz's young side allowed Lyon back into the tie - Alexandre Lacazette netting twice in the space of 76 seconds on the stroke of half-time.
Ajax substitute Donny van de Beek went agonisingly close to putting the tie to bed in the second half when he rattled the woodwork, before Rachid Ghezzal moved Lyon within a goal of forcing extra-time, his header from Maciej Rybus's looping cross deflecting in off Nick Viergever.
Matters were made worse for Viergever when he picked up a second yellow card moments later, but the undermanned visitors clung on for a 5-4 aggregate triumph amid sustained late pressure from Lyon to set up a final against Manchester United on May 24.
Having weathered an early Lyon storm, Ajax had the better of the opening stages - Amin Younes and Hakim Ziyech both failing to make the most of gilt-edged chances.
But the visitors did not have to wait too long to take the lead.
Quick-thinking from Andre Onana freed up Younes to play in Dolberg who - after timing his run to perfection - lofted a sublime finish over the onrushing Lopes.
A dejected Lyon were handed a lifeline just prior to the interval, though - Lacazette thumping home from the penalty spot after going down under pressure from Matthijs De Ligt.
That goal sparked Lyon into life, and they had their second a minute later when Lacazette steered in Nabil Fekir's cross-shot to complete the turnaround.
Ajax's youthful defence was carved open again after the restart, but Mouctar Diakhaby could not apply the finish after connecting with Mathieu Valbuena's free-kick.
Fekir should have turned from provider to goalscorer with 25 minutes remaining, yet Ajax escaped unscathed as the forward skewed over from close range.
Van de Beek was a whisker away from settling the tie with a blistering strike that struck the upright - and Lyon had a third moments later.
Ghezzal's header deflected in wickedly off the unfortunate Viergever, who received his marching orders soon after for a late lunge on Rybus.
But for all their dominance, Lyon could not make the numerical advantage count - Lacazette and Maxwel Cornet both going close as Ajax stood firm to claim a place in a European final for the first time since 1996.