The full-back scored his first goal for the club in somewhat fortuitous fashion - his header hitting the defender, then rebounding back off him before dribbling in - to give the Giallorossi the early advantage in Graz.
But Wolfsberg midfielder Michael Liendl found the top corner from 25 yards early in the second half, as the Austrian side fought to take a draw from the game.
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The result means both sides remain at the top of the group on four points - with Borussia Monchengladbach drawing 1-1 with Istanbul Basaksehir in the night's other game.
On a night of high spirits at the Stadion Liebenau, as Wolfsberg fans celebrated a home European contest, Roma grew into the first half - as both Nicolo Zaniolo and Justin Kluivert threatened the Wolfsberg goal.
But it was ultimately Spinazzola who would break the deadlock - even if he could barely believe his own luck in doing so.
A first-half of few clear-cut chances was followed with a similarly tight and combative second half - only enlivened by Liendl's impressive long-range strike, which beat the otherwise excellent Antonio Mirante.
Roma pushed for a late winner but, when Nikola Kalinic's close-range effort was blocked, they were forced to settle for a draw - a result that nevertheless does little to dampen the hopes of reaching the knockout phase of the competition.
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