But Bayern Munich also served notice they intend to play in the title race despite a sluggish start, whipping Hannover 3-0 away to jump up into fourth spot and delight coach Louis van Gaal.
"We are back to where Bayern should be," said van Gaal. "We have had a very good week," he added in reference to a midweek European win over Maccabi Haifa which left the Bavarians with a chance of overhauling Juventus in their group.
Crowd trouble marred games, though, with police making 85 arrests after violence flared Friday as local rivals Bochum and Cologne faced off. Another 60 arrests followed ahead of Saturday's encounter between rock bottom Hertha Berlin and Eintracht Frankfurt, who won 3-1.
More arrests followed amid scuffles after Hamburg fans brought traffic to a standstill before a draw at Mainz.
Jupp Heynckes' Leverkusen rose above the turmoil to move on to 30 points, three clear of Werder Bremen, who had gone top Saturday after a 2-2 home draw with reigning champions Wolfsburg.
Leverkusen's Kiessling took his league haul to a league best 12, volleying home the first on 22 minutes after ex-Liverpool defender Sami Hyypia headed on.
Swiss star Eren Derdiyok headed the second seven minutes before the break past veteran Stuttgart keeper Jens Lehmann.
Stuttgart coach Markus Babbel was powerless to stem the tide as Toni Kroos picked out Kiessling on the hour for the third.
Lehmann then dragged down the marauding Derdiyok in the box three minutes from time to allow Kiessling the chance to complete his treble from the penalty spot a minute from time as he gave way to Patrick Helmes, returning from injury.
Leverkusen are hoping to stay the course and finally lift the title after several near misses - most notably in 2002 when, with stars such as Michael Ballack in the side, they came second in the league as well as losing the Champions League final and the German Cup final.
Stuttgart in contrast stand second bottom.
In brushing Hannover aside, Louis van Gaal's Bayern emphatically ended a run of three draws to leapfrog Hamburg on goal difference and move fourth on 24 points.
The match was the first at home for Hannover since the recent suicide of their Germany international keeper Robert Enke.
Thomas Müller notched Bayern's opener after 19 minutes, converting a cross from Ivica Olic for his eighth goal of the season.
Olic headed the crucial second two minutes after the break and Mario Gomez put the icing on the cake in injury time as Bayern reduced the gap on Leverkusen to six points.
On Saturday, Werder Bremen had thanked a last-gasp equaliser from Per Mertesacker for their draw with Wolfsburg.
Bosnian sharpshooter Edin Dzeko had opened the scoring three minutes before the break for Wolfsburg, who slid to eighth after Bayern's win.
Portuguese star Hugo Almeida levelled but Dzeko's second five minutes from time looked to have settled it for the visitors until Mertesacker struck in injury time with a header to preserve their 20-match unbeaten sequence.
Schalke had hoped to take advantage of Werder's slip with a win at struggling Borussia Mönchengladbach but instead went down 1-0 with Marco Reus scoring after just five minutes.
The loss was Schalke's first on their travels since last May but Felix Magath's side stay in touch on 25 points in third place.
Hamburg could only manage a disappointing 1-1 home draw with Mainz while Hoffenheim were unable to take advantage of that stalemate as they fell to a 2-1 defeat at home to Dortmund to stay fifth.
At the other end of the table, Hertha Berlin look a solid bet for the drop after a 3-1 home loss to Eintracht Frankfurt - their 11th defeat of the campaign - left the capital club seven points from safety.