League leaders AC Milan beat Genoa later in the day to keep their spot atop the Serie A table.
Chances came and went as Inter looked for the crucial second goal, before Argentine striker Lautaro Martinez nudged one into the net in the 73rd minute to put his side in command.
The hard-fought success moved Inter onto 69 points from 32 matches, but Milan's win over Genoa at San Siro kept them two points ahead of their city rivals. Spezia stay 15th on 33 points.
"I think it was a victory of maturity," Inzaghi told Sky Sport Italia. "We knew it would not be easy. I did not like the first 10 minutes, we gave them too many opportunities with the ball. But it was an important victory for us.
"We are creating a lot of chances, helping us to win the last three games, which is huge for our confidence."
One win in seven Serie A matches before their trip to fellow title hopefuls Juventus two weeks ago had seen Inter fall behind in the race for the Scudetto.
However, after grinding out a 1-0 victory in Turin, Inzaghi's side have not looked back.
Following a couple of early scares in Liguria, Brozovic picked the perfect moment to score his first Serie A goal of the season to edge Inter into the lead.
Inter dominated the second half, and should have added a second through Ivan Perisic and Denzel Dumfries, before Martinez got the faintest of touches to a Perisic cross to increase the visitors' lead.
Maggiore's arrowing strike two minutes from time briefly lifted the home crowd, before Sanchez finished off a swift counter to seal the success.