The French team played with verve throughout the second half on Wednesday but were unable to find the second goal that would have extended their winning streak to three matches.
Playing in Europe's top competition for the first time, the newcomers faced their toughest test yet against the German champions, who had also won their first two matches in the revamped tournament.
Both Leverkusen and Brest missed the chance to join Aston Villa on nine points at the top of the standings.
Not much separated the two sides until a brilliant pass from Jonas Hofmann found Florian Wirtz behind the defence in the 24th minute. The Germany midfielder controlled with a fine touch and finished with a low shot to give Leverkusen the lead. It was his seventh goal in all competitions this season.
Wirtz showed his individual brilliance soon after as he dribbled past three opponents and forced goalkeeper Marco Bizot to make a good save.
Lees-Melou, who recently returned from a broken leg, levelled in the 39th with a right-footed volley from outside the box. It was his first goal since March.
Leverkusen's Granit Xhaka was booked after the final whistle for remonstrating with the referee.
Elsewhere, Celtic struggled but frustrated Atalanta with a 0-0 draw in Italy. Mario Pašalić hit the woodwork for the Italian team.
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