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.
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.
