After a difficult run of results, Spalletti's side responded with a determined performance at the Allianz Stadium, where Alajbegovic broke the deadlock in the 70th minute before Boga added a second seven minutes from time. Napoli's task became harder when Luca Marianucci was sent off for a second booking in the 75th minute.
Spalletti was particularly pleased with the commitment shown by his players, believing they had taken responsibility for rebuilding their relationship with the supporters.
"The players have taken it upon themselves to regain the trust of our fans. They went out there to compete in duels and run for their team-mates and for the team. They won the match deservedly, so congratulations to them," he said.
"If they don't take that first step themselves, after falling as low as we did on Wednesday, it becomes difficult.
"You can make them do anything, but no tactic or system works if you don't win your duels, gain those invisible metres and hold your position higher up the pitch, as we did in the second half."
The Juventus coach also highlighted the difficulties created by his limited options, insisting the squad's injury situation was restricting what he could achieve on the training ground.
"We are 13 players. Sarr couldn't play tonight. People keep shouting that Cambiaso shouldn't be selected, but we don't have anyone else. Cabal may have picked up an injury, and we need to check on McKennie as well. At the moment, we cannot compete in anything unless we get players back."
Alajbegovic's long-range strike gave Juventus the breakthrough, with Boga curling home the second after replacing the goalscorer. Spalletti praised the contribution of his substitutes and the atmosphere created by the home support.
"Napoli took control of the game and we struggled to move across the pitch. Vicario made a great save from Lucca, then gradually we started playing more openly. It was a below-par match for both teams, but the second half went well for us.
"The stadium got involved. That is the real face of the Stadium. Then we needed that great goal from Alaj, who has the ability to shoot from distance. Boga, Zhegrova and Gatti made the difference."
Napoli assistant coach Marco Landucci acknowledged that his side had failed to maintain their first-half performance, but insisted the response must come through work rather than excuses.
"We had a good first half, but Juventus found those two shots in the second half. The difference is made by details. We have to keep quiet and work," he said.
"Juventus deserved to win. It is not enough to play well for one half to win a match. When you lose, you keep quiet and go home."
Landucci also confirmed Giovanni Di Lorenzo had taken a knock, while Kevin De Bruyne had twisted his ankle. Napoli must now assess their availability before turning attention to Wednesday's trip to Porto.