The Bianconeri delivered an improved performance in front of their supporters, with Alajbegovic breaking the deadlock in the 70th minute before Boga doubled their advantage seven minutes from time.
Napoli's hopes of mounting a comeback suffered a major blow when Luca Marianucci received a second yellow card in the 75th minute, leaving the visitors with 10 players for the closing stages.
The result provided some relief for Spalletti following Juventus' inconsistent run of form, while Napoli suffered their fourth league defeat of the campaign.
Juventus and Napoli struggle to break the deadlock
Both sides approached the contest cautiously, with Juventus looking to take control while Napoli attempted to exploit spaces on the counter-attack.
Napoli created the first significant opportunity in the seventh minute when Lorenzo Lucca connected with a cross, but Juventus goalkeeper Guglielmo Vicario reacted brilliantly to keep out his effort.
Juventus gradually grew into the game and came close to opening the scoring through Juan Cabal, whose header from a corner struck the post.
Napoli responded almost immediately, but Noa Lang wasted a promising opportunity by firing over the crossbar from a good position.
Alajbegovic also missed a chance to put the hosts ahead, while Alex Meret produced an important save to deny the young Bosnian shortly before half-time.
Despite the opportunities at both ends, the first half ended goalless.
Alajbegovic breaks the deadlock with stunning strike
Juventus continued to search for an opening after the restart, with Spalletti introducing tactical changes to increase his side's attacking threat.
The breakthrough finally arrived in the 70th minute following an excellent piece of individual play from substitute Edon Zhegrova.
The Kosovo international beat Amir Rrahmani before finding Alajbegovic, who took advantage of the space outside the penalty area and unleashed a powerful left-footed effort into the far corner.
Meret had no chance of stopping the spectacular strike as the Allianz Stadium erupted in celebration.
The goal was Alajbegovic's first in Serie A and gave Juventus the advantage they had been searching for throughout the contest.
Marianucci sent off as Boga seals victory
Napoli's chances of recovering from the setback diminished five minutes later when Marianucci received his second yellow card for a foul on Cabal.
The dismissal left the visitors with 10 men and gave Juventus greater control of the closing stages.
Boga, who had replaced Alajbegovic, then put the result beyond doubt in the 87th minute.
After receiving the ball on the left, the forward cut inside and curled a superb right-footed effort into the far corner, leaving Meret beaten for a second time.
The goal completed an impressive evening for Juventus, who held on to secure a deserved 2-0 victory.
Juventus ease pressure on Spalletti
The result was an important response from Juventus after a difficult period that had increased scrutiny on Spalletti.
The victory lifted the Bianconeri to seventh place with 17 points from 10 matches, while Napoli remained eighth with 14 points.
For Napoli, the defeat represented another setback in a challenging start to their league campaign.
Juventus will now turn their attention to their upcoming European assignment before returning to Serie A action against Fiorentina, while Napoli prepare for their next Champions League fixture against Porto.