Walter Mazzarri's side are on 47 points and have leapfrogged Fiorentina, who host Chievo Verona on Sunday, after Palacio's 14th minute tap-in was followed just past the hour by a messy strike from Jonathan who needed three attempts to score.
Hellas stay seventh on 40 points as they continue to battle for a Europa League spot despite two wins in eight games since selling Brazilian playmaker Jorginho to Napoli in mid-January.
Jonathan said: "This is a very important win, because they (Hellas) are close behind us in the fight for the Europa League. We've got to keep an eye on Parma (sixth on 43 points) as well.
"I'm really happy for how we played, and tomorrow I'm going to be supporting against Fiorentina."
What was at one point looking to be a disastrous season for Inter, with wins hard to come by and the fans railing against new president Erick Thorhir, has turned around in recent weeks.
They are unbeaten in six games and stand eight points behind third-placed Napoli, who visit Torino on Monday.
"I've tried to drill into these players from the start that they must go for the win from the very beginning and maintain the same mentality no matter where they are playing," said Mazzarri.
"That means winning back possession as quickly as possible and seeking the victory every time. You can see that in recent outings the team has indeed taken that approach."
Argentina international Palacio, who signed a two-year extension to his contract on Friday, got his 13th league goal this season after sparkling wing play from Jonathan.
Inter were in control from the start and the could have won by more, with Mauro Icardi hitting the bar with a sixth-minute header and Hernanes being similarly denied after a fantastic free-kick from near the sideline 10 minutes after the break.
They took a deserved lead through Palacio but Jonathan made the goal with sublime skill, the 28-year-old shimmying past Michelangelo Albertazzi with two stepovers before rolling the ball across for the Argentine to slot home from close range.
The Brazilian wing-back sealed the points in less elegant style, latching on to Hernanes' through ball and forcing a smart save from Rafael before scuffing the rebound against defender Maietta Domenico and finally netting from the second rebound.
Goalkeeper Samir Handanovic had a quiet night but was at his best to deny Verona an added-time consolation with a superb double save from Iturbe and Romulo on an almost perfect night for Inter.
Runaway league leaders Juventus will look to move 17 points clear with a win at Genoa on Sunday before second-placed AS Roma host Udinese on Monday.
- Sports & Recreation
- Chievo Verona
- Europa League
- Walter Mazzarri