Midfielder Ianis Hagi fired in from outside the box after seven minutes for Rangers, and the rattled visitors then struggled to keep at bay the relentless Light Blues, who were shorn of several key players.
Brendan Rodgers' champions, without the backing of away supporters, saw a Kyogo Furuhashi goal chalked off for offside early in the second half before Gers defender Robin Propper bundled in a second in the 65th minute.
Substitute Danilo then knocked in a third in the 81st minute soon after climbing off the bench.
Clement will have bragging rights from the Glasgow derby for the first time since becoming Gers boss in October 2023 and stave off talk of the sack which has dominated the narrative in recent weeks.
Rangers are still 11 points behind leaders Celtic in the title race and their hopes still remain very unlikely, but the psychological boost of the victory was clear by the ecstatic celebrations at the final whistle.
More upheaval around Rangers preparations seemed to be a portent of another difficult afternoon against their great rivals.
After dropping five points in their previous two games, Rangers went into the game lacking any momentum and were decimated defensively.
Goalkeeper and vice-captain Jack Butland had to pull out on New Year's Day with a leg issue, joining skipper James Tavernier and fellow defenders Leon Balogun, John Souttar and Neraysho Kasanwirjo on the sidelines.
Liam Kelly, in his second spell at Ibrox but who only made his debut on December 21, deputised for Butland, while left-back Ridvan Yilmaz started on the other side of the pitch as Tavernier's replacement and Nico Raskin was named captain.
There was much less drama around Celtic, who emerged from 2024 with only two defeats in the year - Borussia Dortmund and Hearts - and Callum McGregor, left-back Greg Taylor and forward Daizen Maeda all returned to starting line-up.
But it was to be Rangers' day, with the only downside for the home team coming late on when Celtic sub Arne Engels went down near the corner flag after appearing to be struck by an object, before carrying on after treatment.
