Leverkusen stalemate dents Union's Bundesliga title hopes
Unbeaten at home in the league this season, Union's best chance came midway through the second half when Sheraldo Becker hit a Jerome Roussillon cross wide of the post.
A win would have taken Union three points behind first-placed Borussia Dortmund in just their fourth season of top division football.
Despite the draw, Union remain third and on course for a maiden Champions League placing with four matchdays remaining.
Calling Leverkusen, who are unbeaten in 14, "the most in-form team in the league", Union manager Urs Fischer praised his side's "intense, great performance" in "neutralising" the visitors.
The stalemate also hurt Leverkusen's narrow chances of qualifying for the Champions League, with the Xabi Alonso-coached side now eight points outside the top four.
"Union has not lost here yet this season, and we could sense why" said Alonso.
Elsewhere, Freiburg solidified their chances of following Union to a first-ever Champions League qualification, a second-half Ritsu Doan tap-in taking them to a 1-0 win away at Cologne.
The win took Freiburg to fourth, behind Union on goal difference, two clear of RB Leipzig who also won 1-0 on Saturday.
Leipzig beat Hoffenheim thanks to Christopher Nkunku's first league goal since November.
Nkunku, handed his first start since returning from injury, collected a pass from Emil Forsberg and sent a left-footed finish into the right corner of the goal, chalking up his 13th goal in 21 league matches this season.
Leipzig, who have made the top four in five of their six seasons in the top division, are fifth, six points ahead of Leverkusen.
Stuttgart leapfrogged Bochum out of the relegation places with a 2-1 home win over a ten-man Borussia Moenchengladbach.
The home side took the lead thanks to a classy Serhou Guirassy back-heel finish after 22 minutes.
Gladbach's Julian Weigl converted a penalty with 22 minutes remaining but Stuttgart were awarded a spot kick of their own five minutes later when Ko Itakura was shown a straight red for bringing down Tiago Tomas.
Stuttgart's Tanguy Coulibaly converted the penalty to give his side a welcome boost ahead of Wednesday's German Cup semi-final against Eintracht Frankfurt.
Frankfurt drew 1-1 at home against Augsburg, extending their winless run to nine matches and likely extinguishing any slim hopes the Europa League holders have of a return to Europe.
In Saturday's late game, relegation-threatened Schalke can lift themselves to within one point of safety if they defeat Werder Bremen.
