It was a big blow for the Australian as Pierre Gasly delivered the shock of the season by taking the first pole position of his career for Sunday's race.
Piastri slowed to allow Lawson's teammate Max Verstappen through but delayed the New Zealander - a Red Bull substitute - when he slotted in ahead of him at the first chicane during the closing stages of Q2.
After hearing from both drivers, Red Bull and McLaren team representatives, and reviewing the range of data available to them, the stewarding panel settled on a grid drop for the Australian.
That meant Piastri - currently seventh in the standings - fell from third to sixth on the grid, promoting Ferrari pair Charles Leclerc and Lewis Hamilton, as well as the Red Bull of Lawson's team mate, Max.
George Russell, widely expected to claim a much-needed pole in his world championship quest, had to settle for second on Saturday, pipped to top spot by Gasly by just 0.060 seconds.
Speaking before the penalty was applied, Piastri said: "It was a tricky situation to manage. I was coming out of the pits and got told that Lawson and Verstappen were coming, but I kind of had to jump on the brakes pretty heavily to get out of Max's way, and then I was just stuck in a very awkward spot with Liam."
Piastri added: "The speed difference coming out of the pits is so big here, so I don't really know where he came from."
Gasly's improbable triumph arrived a day on from his Alpine boss Flavio Briatore accusing an FIA judge of having a "working relationship" with McLaren after the British team's appeal led to the Frenchman being stripped of his podium from June's Monaco Grand Prix.
Championship leader Kimi Antonelli took seventh but will start his home race from the back of the grid as he serves an engine penalty.
World champion Lando Norris arrived in Italy with consecutive victories under his belt but looks almost certain to miss out on a hat-trick after he could manage only ninth.
With Antonelli starting Sunday's 53-lap race from the rear, Russell knew this event would hand him a golden chance to eat into his Mercedes teammate's 59-point championship lead.
But the Englishman will have to see off Gasly's unexpected challenge to claim a full quota of points in what could prove a pivotal contest in the title race with 11 rounds still to go.
Hamilton spoke about his dream of winning in Monza for a record sixth time and a first for Ferrari in the build-up to the Italian team's home event.
But the seven-time world champion endured a difficult afternoon.
At one stage, Hamilton's presence in Q3 appeared in grave danger after he could set only the 10th quickest time. However, his blushes were spared by just one- thousandth of a second.
It was an unsatisfactory afternoon for Red Bull with Verstappen – who claimed his car was "broken" and "undriveable" – three-tenths off the pace.
But there were no such concerns for Gasly as he flourished in his Alpine.
"I have been waiting for that one," yelled the Frenchman – whose sole victory of his Formula One life arrived in Monza six years ago – over the radio.
Fernando Alonso qualified 21st, with only Aston Martin teammate Lance Stroll slower.
With agencies
8°C