Two goals from Mauro Icardi and a fine strike from substitute Marcelo Brozovic sent the Nerazzurri to the Serie A summit, but it was far from comfortable for Spalletti's side.
The hosts could have been at least two goals to the good before Icardi's first just before the half-hour mark gave Inter an underserved lead.
Leonardo Pavoletti made it 2-1 with 20 minutes remaining to set up a potentially tense finale, but Icardi's second put the game to bed.
Spalletti was full of praise for Cagliari's high-energy performance and concedes he had to change his approach in order to find a way into the game.
"This was the performance of a strong side, because we started with some difficulties," he told Mediaset Premium.
| FT - Cagliari 1-3 Inter— Inter (@Inter_en) November 25, 2017
Job done! @MauroIcardi and #Brozovic with the goals as we do the business in Sardinia. Good stuff lads! That's points now!
#CagliariInter #ForzaInter pic.twitter.com/PuxgJxpaFY
"We tried to press them higher, as we were reacting as individuals rather than as units, and Cagliari were flinging themselves into the fire to get to these loose balls first.
"We had some difficulties creating moves from the back, as Cagliari were pressing us so hard it was violent. They were rushing at us at full force the second we got the ball.
"We had to modify a few things as Cagliari had an excellent start and we suffered under pressure, the way a great team ought to.
"I wanted to change our shape, so we basically mirrored Cagliari's tactics, but it was an issue of getting to grips with the distances and movements. In the second half, we went back to a four-man defence and played more of our football."
Icardi, whose brace took his league tally for the season to 15, believes that Inter did not start with the right attitude — something that changed after his first goal.
"The attitude we had in the first half was a bit of a problem, then my goal calmed Cagliari down a bit," he added. "They had a really good game.
"We do our work, focus on our results and winning our games. The other teams know we are there, we’re pressing them and will enjoy tonight as the Serie A leaders.
"We've been working well for a couple of years, doing great things and we can see the fruits of our labour. I try to make sure I am ready when the ball comes my way. I got another brace and I'm enjoying my football."