An own goal from Francesco Vicari had looked set to give the visitors their first win in seven Serie A matches until Alberto Paloschi's 90th-minute header ensured the spoils were shared at the Stadio Paolo Mazza.
The setback for Inter, who last won in the league on December 3, means Roma can move level on points with them if they beat Sampdoria, while the gap to the top three will stand at five points if Lazio win at AC Milan.
Spalletti was frustrated to see the same sort of performance Inter have produced in recent weeks and has called on his players to show "a strong reaction".
"When you concede goals like that, you're sorry," he told Mediaset Premium. "The bitter taste stays in your mouth. You work harder but nothing changes.
"It's always the same thing. I don't see any improvement. Even when we were winning games, there were these blackouts where we lost simple balls and at other times didn't fight.
"We need a balance of behaviour and a consistency of mentality."
He added in a news conference: "The team work well and keep their balance on the pitch, but then we lose duels in trivial ways. We have to work better and find more solutions, without becoming timid and fearful.
4 - Inter have drawn in 4 Serie A matches in a row for the first time since November 2004. Pause. #SPALInter— OptaPaolo (@OptaPaolo) January 28, 2018
"We're going through a moment that doesn't help us to play calmly. It feeds the tension. Our self-esteem is so severely tested.
"We need a strong reaction."