Leverkusen are now unbeaten in five Bundesliga matches to sit 10 points clear of Schalke, and they will be assured of a top-three finish if the Gelsenkirchen club fail to take a point against Hamburg tomorrow.
Bremen, however, are without a victory in 10 and remain perilously close to the relegation zone with just three matches to escape.
The home side had the opening chance after four minutes as Daniel Carvajal crossed into the centre of the area, where Kiessling headed wide.
However, they grabbed the lead in the 35th minute from the spot, with Kiessling slotting the ball into the bottom right corner just out of the reach of goalkeeper Sebastian Mielitz, who had fouled Sidney Sam in the box to concede the penalty.
The goal was Kiessling's 22nd of the campaign and moved him level with the club's record top goalscorer in a season, Ulf Kirsten, who notched 22 goals in both 1996-97 and 1997-98.
Leverkusen continued to push forward for a second goal but Philipp Wollscheid headed wide from six yards out seven minutes into the second half.
Andre Schurrle's shot from distance then went just over the crossbar for the hosts after an hour.
Bremen finally had a chance to pull level after 69 minutes but Nils Petersen headed wide of the right post.
And the visitors came close again three minutes later as Zlatko Junuzovic pounced on a mistake by Wollscheid, but Leverkusen keeper Bernd Leno made a fine save.
Lars Bender had the opportunity to settle the match for good for Leverkusen on a counter attack in the final minute, but he pushed his shot past the left post.