Palacio, a 10th-minute replacement after a serious-looking injury to Diego Milito, scored 10 minutes after that and added a late second.
Those two goals, coupled with a clean sheet, meant next week's trip to Transylvania for the second leg should hold few terrors for the Serie A side.
Cluj's Rui Pedro and Roberto Maah shot wide, the latter after Mario Camora's free-kick was only half-cleared, before Inter were forced into their early change.
And after Esteban Cambiasso volleyed wide from close range and Mateo Kovacic's drive was saved, Palacio opened the scoring in the 20th minute.
The substitute latched on to Fredy Guarin's through-ball and finished low inside the near post from the right side of the penalty area.
Gabriel Muresan fired in two poor long-range efforts as Cluj sought a response, before Palacio forced a sharp save from keeper Jorge Felgueiras at the other end.
Felgueiras saved again from Cambiasso before Cluj suffered a first-half injury of their own, Nicolas Godemeche replaced by Vasile Maftei.
Guarin was just off target with an overhead kick but Cluj could have equalised before half-time, midfielder Ionut Rada curling one free-kick over the bar before a second was brilliantly saved by Samir Handanovic. Inter's Mateo Kovacic sent a 20-yard effort wide in between times.
Antonio Cassano shot wide and Guarin dumped a free-kick into the wall early in the second half, before the lively Palacio had a curling shot blocked by Ricardo Cadu.
Camora's free-kick from out wide caused problems for Handanovic but Inter's goal was never seriously threatened despite Cluj boss Paulo Sergio using both his remaining substitutions.
And the hosts extended their advantage when Palacio, having sent one shot high over the bar, notched his second of the night with four minutes remaining.
The Argentinian was sent clear by Kovacic's long pass and deftly clipped the ball over Felgueiras.
Guarin shot wide as the match entered injury time but Andrea Stramaccioni's side had done what they had to.