The 36-time Italian champions had only claimed two points from their first four games this term, and were sitting in the relegation zone heading into Wednesday’s game.
Their maiden victory did not come easy, coughing up an early lead from Moise Kean’s low 28th minute strike, with the hosts hitting the front early in the second half after Emmanuel Gyasi and Janis Antiste goals.
But Federico Chiesa forced a 66th-minute equaliser before Matthijs de Ligt slammed home a loose ball for the lead in the 72nd minute.
“The most important thing was we won and I’m pleased because we had to dig in to get it,” Allegri told reporters after the game.
“Spezia play good football. We weren’t able to go 2-0 up but after Spezia took the lead we responded well.
“We were a bit tense because we’ve lost or drawn games in quite incredible fashion but we managed to limit the danger today.”
Allegri conceded there was plenty of room for improvement for Juve, who remain down the wrong end of the table, sitting 13th with five points.
“We’ll savour this win now and from tomorrow we’ll concentrate on trying to win our first game at home,” Allegri said.
“We need to hone our focus levels and enjoy ourselves a bit more too but we defended the right way today. It’s hard to maintain the same line-up with so many games back to back but one thing we do need to work on is improving our ball circulation.”
AC Milan kept up their strong start to the campaign with a 2-0 home win over Venezia after second-half goals from Brahim Diaz and Theo Hernandez.
Milan are second with 13 points from five games, with Inter Milan top after beating Fiorentina 3-1 on Tuesday.