Alonso and Button ended Red Bull's run of 17 successive races with at least one driver on the front row and was only the second time this season that the championship-leading team had not taken pole.
Under the watchful eye of Ferrari president Luca di Montezemolo, who punched the air in jubilation from the pit wall at his team's first pole at their home circuit since 2004, Alonso was quickest when it mattered.
"Tomorrow we need a podium, minimum, to remain in the fight," said the double world champion, who will be making his first race appearance at Monza as a Ferrari driver after joining from Renault last year.
"There is not a big pressure on us... but we cannot afford another DNF (did not finish) and a bad result," added Alonso, who has won twice already this season but is fifth overall and 41 points off the lead.
Button, back on the front row for the first time since Turkey last June, also needs to claw back points from Hamilton.
"A podium would be good but we have to go for the victory," he said.
"It's going to be an interesting race with cars running such different downforce levels."
Button went with a very different setting to Hamilton, who decided not to use the 'F-duct' device that stalls the rear wing and improves straight line speed.
The device offers less of an advantage on Monza's long straights because cars have special rear wings designed for maximum aerodynamic effect.
McLaren arrived as favourites at the fastest circuit on the calendar, while Red Bull had expected to struggle, and Button said he was surprised by Ferrari's pace.
Alonso's F1 title bid on the line at Italian GP
"I personally didn't expect the Ferraris to be as quick as they are," he said.
Hamilton, the only other non-Red Bull driver to take a pole position this season, hoped to recover the situation on Sunday.
"Hopefully we should be quite fast on the straights compared to the others," he told reporters.
"I didn't have enough downforce, really not enough. I was sliding around a lot and I couldn't pick up the pace. Jenson did a great job to put himself second."