Adam Szalai had put Mainz ahead two minutes before the interval, but the visitors were reduced to 10 men in first-half stoppage time when Shawn Parker was sent off for a bad tackle on Augsburg substitute Michael Parkhurst.
The hosts soon capitalised on their extra man, Molders restoring parity 12 minutes after the restart, but they could not force a winner.
The result marked Augsburg's sixth point in four unbeaten matches since the winter break and lifted them to 15 points - one behind third-bottom Hoffenheim - while Mainz stayed in European contention in fifth spot.
Augsburg goalkeeper Alex Manninger was called into action as early as the sixth minute after home defender Jan-Ingwer Callsen-Bracker deflected Szalai's blast towards his own goal.
The hosts went close to taking the lead 10 minutes later, but Marcel de Jong's deflected shot flew inches wide.
Parker thought he had put Mainz ahead midway through the first period from Julian Baumgartlinger's through-ball, but he was ruled offside.
The away team did take the lead in the 43rd minute, though, when Szalai lashed home from close range after good work by Junior Diaz on the left.
Andreas Ivanschitz had the chance to double Mainz's lead in the 45th minute but fired wide, and the half ended on a sour note moments later when Parker was dismissed for a dangerous high challenge on Parkhurst.
Augsburg drew level before the hour, Molders directing his header home from De Jong's long cross.
The 11 men were in the ascendancy thereafter but could not press home their advantage. Parkhurst despatched his shot into the side-netting from their best opportunity in the 67th minute.
Mainz had half a chance to nick the win eight minutes from time, but substitute Marcel Risse drove his long-range effort narrowly wide.