Inter, who beat Partizan Belgrade 3-1 in Serbia with all the goals coming in the second half, were joined in the next phase by Bayer Leverkusen, Rubin Kazan, Metalist Kharkiv, Hannover 96 and Olympique Lyon.
Russian side Kazan moved to the top of Group H, level on 10 points with Inter, after a 1-0 win at Azerbaijan's Neftchi Baku thanks to Vladimir Dyadyun's 16th minute strike.
Leverkusen and Metalist were also level on 10 points in Group K, the Germans too good for Rapid Vienna with a 3-0 home win while their Ukrainian rivals beat Norway's Rosenborg 3-1.
Palacio's double for Inter and a third from Fredy Guarin late on saw off Partizan, who netted a consolation goal in injury time through Nemanja Tomic.
Jens Hegeler's fourth-minute strike got Leverkusen off to the perfect start and goals after the break from Andre Schuerrle then Manuel Friedrich settled the outcome.
Hannover's last-gasp win came courtesy of an injury-time Szabolcs Huszti penalty that took the Germans to 10 points in Group L. Levante held Twente Enschede 0-0 in the same group and are on seven points.
Lyon maintained their perfect record with a 3-2 win at last year's finalists Athletic Bilbao to streak ahead of Group I rivals Sparta Prague, still in second after a 1-1 draw away to Israel's Hapoel Ironi Kiryat Shmona.
Czech side Viktoria Plzen took advantage of Atletico's slip-up and joined them at the top of Group B after easing to a 4-0 home win against Hapoel Tel Aviv.
With one eye on Sunday's Premier League clash at Chelsea, Liverpool manager Brendan Rodgers fielded a makeshift side who were undone by the lanky 22-year old striker.
