Santander's header gave new coach Sinisa Mihajlovic the perfect start as his relegation-threatened side earned their first win in over four months.
But Spalletti's Inter were booed off the pitch at the San Siro after their third-place cushion was cut to four points on city rivals AC Milan, who occupy the final Champions League berth, with Roma a further point back in fifth.
Paraguayan forward Santander broke through after 33 minutes with his header enough to seal just a third win this season and first since September 30.
Inter are still searching for their first win of 2019, being held to a goalless draw by Sassuolo and losing 1-0 to Torino in Serie A before exiting the Coppa Italia on penalties to Lazio in the San Siro this week.
“Unfortunately when we have to fight or to cope with difficulties we lack something, that bit of dash that can make you make a difference,” said Spalletti.
“In any case, I repeat, I don't feel under scrutiny. The team is still behind me, I'm convinced,” added the former Roma and Zenit St Petersburg coach.
The damage could have been worse but for a solid performance from goalkeeper Samir Handanovic, with captain Mauro Icardi proving wasteful, extending his goal drought in Serie A to six games.
Bologna keeper Lukasz Skorupski denied Andrea Ranocchia a late equaliser as the northern side reignited their push for top-flight survival.
“I couldn't have dreamed of a better start,” said Serb Mihajlovic.
“This proves that if you are brave you can do anything – you can beat Inter at the San Siro.”
- Italian team Inter spurred on by return to 'Disneyland of football' in clash with Tottenham
- Milan coach hits back as Italy's deputy PM slams his game plan
- Five things we learned in Serie A
'Slap in the face'
At the Stadio Olimpico, Roma eased the pressure on their coach Eusebio Di Francesco who was still stunned by a 7-1 Coppa Italia whipping by Fiorentina.
AC Milan's Lucas Paqueta combined to set up fellow new boy Krzysztof Piatek to tap in after 26 minutes for his third goal in two outings.
But Roma piled on the pressure with AC Milan saved by goalkeeper Gianluigi Donnarumma's heroics denying Patrik Schick and Edin Dzeko before the break.
Just after the interval Nicolo Zaniolo equalised for the hosts who were boosted by the return of veteran midfielder Daniele De Rossi.
Donnarumma again came to the rescue to frustrate Dzeko, while a searing Lorenzo Pellegrini header rattled the post, with Roma goalkeeper Robin Olsen denying Diego Laxalt late.
“It's a pity we need a slap in the face before waking up,” said Di Francesco. “This was a positive response to Florence, but could never wipe out that game, even if we had won.
“The lads showed character and I hope it is yet another starting point.”
“Roma probably deserved a little more, but I really liked our second half performance,” said Milan coach Gennaro Gattuso.
Both Milan and Roma edge closer to Inter but Atalanta and Lazio have a chance on Monday to close in on the top four against Cagliari and Frosinone respectively.
Earlier Fiorentina battled back to share a point in a 1-1 draw with Udinese, as the Tuscany side sit five points off the Champions League places.
The match was halted at the 13th minute as the fans remembered former Fiorentina captain Davide Astori, who died before last year's game in Udine.
Elsewhere Paraguayan striker Toni Sanabria scored his second goal in as many games for Genoa since his loan move from Real Betis, in a 1-1 draw with Sassuolo.