- Federico Chiesa (68')
The Old Lady came seventh with 62 points and qualified for the Conference League, while Atalanta (64) and this year's Europa League runners-up Roma (63) ended fifth and sixth respectively.
Juve dropped to seventh last month after being docked 10 points by an Italian soccer court over their transfer dealings.
Allegri described the campaign as a complicated one for Juve, highlighting how they were initially docked 15 points, with that decision suspended following an appeal.
They would have finished third without the subsequent 10-point penalty that took them out of the Champions League spots.
"The 10-point penalty was not our fault, so I took the team to third place on the field and nobody can take that away from me," Allegri told Sky Sport Italia.
"This was a difficult year, but also a wonderful one. The [Udinese] game was largely irrelevant. I just want to thank all those who worked with me at the Continassa [training ground]."
Asked about his future, Allegri said: "I'm not discussing any contract termination, I still have two years on the deal."
With little at stake for Udinese, the hosts took a defensive approach in the first half as Juventus dominated possession.
Juve came close to scoring near the half-hour mark when Leonardo Bonucci's powerful header struck the bar following a flick-on from a corner.
Chiesa broke the deadlock in the 68th minute by expertly sending the ball into the bottom corner from 15 metres.
- Marco Guida