In a sparkling display of attacking football, Kiessling scored either side of an Eren Derdiyok goal before drilling in a late penalty to leave the relegation-threatened visitors one place off the bottom on 11 points.
"Today we showed what kind of football we are capable of playing," said Kiessling. "We did not concede anything and this is how we are playing at the moment."
Leverkusen, undefeated this season, are on 30 points, three ahead of Werder Bremen, who salvaged a 2-2 draw against champions VfL Wolfsburg with a stoppage-time goal on Saturday.
Bayern Munich climbed up to fourth place on 24 points, after beating Hanover 96 3-0 with Croatia striker Ivica Olic setting up the first for Thomas Mueller and heading in the second. Mario Gomez added a third goal in stoppage time.
Stuttgart, who saw some of the pressure of a dismal season lifted in mid-week with a win over Rangers in the Champions League, were sluggish from the start, allowing Leverkusen to threaten early on with a powerful Kiessling volley.
The Germany striker, battling for a spot in next year's World Cup squad, turned provider in the 20th minute, cutting into the box for Toni Kroos to hit the post.
Kiessling scored two minutes later when Kroos sailed a deep free kick into the box with Sami Hyypia heading the ball on for the 25-year-old to fire home.
Kroos, the gifted 19-year-old on loan from Bayern Munich, came close on 32 minutes but his low drive hit the post again.
Swiss Derdiyok fared better seven minutes later, cleverly beating the offside trap to head in a pinpoint cross from Kroos as the Stuttgart defence looked in complete disarray.
Kiessling and Kroos picked up where they had left off in the second half, with the two combining well and the Bundesliga's top striker slotting home Leverkusen's third goal in the 59th.
The hosts then took their foot off the gas with Kiessling grabbing his 12th goal of the season four minutes from time, converting a penalty awarded for a foul on Derdiyok by keeper Jens Lehmann.
"We just played crap from start to finish," Germany international Serdar Tasci told reporters.
The defeat put the pressure back on Stuttgart coach Markus Babbel, with fans long calling for his resignation.
"Everyone must have realised by now that we have arrived at the relegation battle," said Babbel. "We now need to start the coming week fully focused."
