Andre Hahn gave the hosts a deserved lead in the 27th minute, but a defensive Borussia replied through Max Kruse six minutes later.
Markus Weinzierl's men continued to call all the shots, but it was a counter-attacking Gladbach who took the lead with substitute Branimir Hrgota on target in the 72nd minute.
That goal seemed good enough to earn the visitors all three points, but Milik had other ideas as he opened his Bundesliga account with an 88th-minute equaliser.
Augsburg showed plenty of attacking intent early on and they almost made the breakthrough with Halil Altintop striking the outside of the post early on.
Gladbach assumed their customary cautious approach, appearing to invite pressure on them and hope for a chance to make a breakthrough on the counter-attack.
Their first opportunity almost resulted in a goal, but Juan Arango shot against the crossbar with only Alexander Manninger to beat.
Augsburg continued to attack and they were rewarded in the 27th minute when Hahn broke through and provided a cool finish into the far, left-hand corner, which betrayed the fact it was his first ever Bundesliga goal.
Their lead lasted only six minutes before Kruse's fourth goal of the season brought the Foals level.
He looked even more assured than Hahn as he switched the ball from his left to his right foot before stroking it into the bottom left-hand corner.
Undeterred, Augsburg continued to do all of the running as Gladbach invited them forwards.
Gladbach coach Lucien Favre then sensed an opportunity and sent Hrgota on for Patrick Herrmann, seeing the move pay off within minutes as the substitute put the visitors ahead in the 71st minute with a fine left-footed volley past Manninger.
Gladbach continued to defend, this time with something to hold on to, but they fell two minutes short of their first away win as Milik struck from a tight angle to earn the hosts a deserved point on the balance of play.