Walter Mazzarri's men were denied by the crossbar in the early stages, but Palacio, who put pen to paper on the new deal on Friday, soon tapped home from close range to give Inter a deserved lead.
The visitors at Stadio Marc'Antonio Bentegodi seemed happy enough to sit on that one-goal advantage, with clear-cut opportunities sparse for much of the Serie A encounter, but Jonathan found the back of the net shortly after the hour mark to make sure of the three points.
Inter's run of six games unbeaten has now seen them climb to fourth and two points clear of Fiorentina, who face Chievo on Sunday, while Verona – without a win in six home matches – lost further ground in the chase for a UEFA Europa League berth.
The late withdrawal of Yuto Nagatomo through injury saw Mazzarri hand a first Inter start to Danilo D'Ambrosio, and the visitors went agonisingly close to breaking the deadlock in the opening five minutes as Mauro Icardi headed a looping header against the bar.
Icardi then had appeals for a penalty turned down shortly afterwards as he hit the deck after going shoulder-to-shoulder with Verona defender Vangelis Moras on his way into the Verona area, but Inter did edge ahead after 13 minutes.
Jonathan carried the ball well down the right-hand side before drilling in a low cross for Palacio, who tapped home from barely three yards out.
Verona had a penalty shout of their own just before the half-hour mark as Luca Toni went down theatrically under the challenge of Andrea Ranocchia, but referee Luca Banti was having none of it.
Rafael had to pull off a point-blank save when Fredy Guarin was sent through on goal by Jonathan after 33 minutes, with Verona struggling to test Samir Handanovic at the other end.
Hernanes tried his luck from distance six minutes into the second half, but sent his effort straight into the arms of a grateful Rafael before Inter had the bar rattling for a second time.
Michelangelo Albertazzi and Jonathan had a coming together on the right-hand side of the penalty area – Hernanes being denied by the woodwork after Inter were awarded the free-kick.
Jonathan, who was involved in most of Inter's attacking moves, added his own name to the scoresheet in the 63rd minute, reacting well after Rafael had saved his initial effort to follow up and smash home from close range.
Moras had a glorious opportunity to reduce the deficit nine minutes later, but headed just over the crossbar from an unmarked position in the Inter box.
Hernanes almost extended the Inter lead 12 minutes from time, but his dipping effort whistled just over the target.
Handanovic was forced into a double diving save in the closing stages as Juan Iturbe and Romulo both tested the Slovenian.
