Gladbach captain Daems converted the 27th minute spot kick after Augsburg defender Kevin Vogt was sent off for a foul on Peniel Mlapa.
Togo-born Mlapa could have added another goal for the home side minutes later after a perfect pass from Mike Hanke but the Germany Under-21 international hit the post.
Gladbach missed several golden opportunities, with Hanke hitting the post in the 73rd, and were almost made to pay when Augsburg's Knowledge Musona forced a superb save from goalkeeper Marc-Andre ter Stegen in the 90th minute.
Gladbach moved up to 44 points from 30 games, one point behind fifth-placed Freiburg and two off Schalke 04 in fourth.
Augsburg, who had won four of their previous seven games to move out of the bottom two in recent weeks, stay in the relegation playoff spot with 27 points, three above Hoffenheim.
Bayern Munich, who have already secured the title, will be looking to equal Borussia Dortmund's record haul of 81 points in a season if they win at Hanover on Saturday when Dortmund, 20 points adrift in second place, host Mainz.