Serie A: Cesena 1 Milan 1
Filippo Inzaghi's men controlled the game for large periods, but were undone early on when veteran goalkeeper Abbiati fumbled a weak shot straight into the path of Davide Succi, who gleefully accepted the gift.
Rami rescued his team with a towering header after 19 minutes on an otherwise frustrating afternoon for Milan that was compounded when Cristian Zapata was shown a straight red for bringing down substitute Gregoire Defrel.
Nicola Leali impressed in the Cesena goal, thwarting Mattia De Sciglio in the second half, while for Cesena, Franco Brienza fired over a free-kick from a good position late on.
But neither side could find a winner and it is now three games without a victory for a stuttering Milan side, while newly promoted Cesena will take great heart from the result.
Inzaghi was forced to change the side that battled from two goals down to draw 2-2 at Empoli on Tuesday as the injured Marco van Ginkel (ankle ligaments) made way for Andrea Poli.
Pierpaolo Bisoli, Inzaghi's opposite number, welcomed back Massimo Volta and Brienza, who both missed the 3-0 defeat at Juventus in midweek with flu.
Fernando Torres forced Leali into a ninth-minute save with a header at the near post, but the visitors fell behind a minute later with a goal that Abbiati will want to forget.
Cesena's Guido Marilungo found space in the box and, although he got a shot away, the effort was weak. But the Milan goalkeeper somehow fumbled the ball into the path of Succi and he duly tapped in his first of the season.
Milan were rattled and could have conceded a penalty moments later through a clumsy Rami challenge on Succi before they found their feet and Keisuke Honda's free-kick had to be pushed over by Leali after 17 minutes.
Cesena did not heed the warning and Milan levelled moments later when Honda's superb corner was met by Rami, who headed past the helpless Leali into the far corner.
The goal seemed to settle the visitors. Jeremy Menez picked out Giacomo Bonaventura in the area after 30 minutes, but he fired over.
Both sides had chances early in the second half. First, the ball sat up perfectly for Manuel Coppola and his effort went just wide of the post with Abbiati scrambling.
And at the other end, De Sciglio cut inside from the left and left fly with a low effort that Leali tipped behind.
For all their possession, Milan struggled to break down a resilient Cesena defence and when Bonaventura set up Torres after 62 minutes, the Spaniard headed over.
Milan had to play the final 17 minutes with 10 men after Zapata hauled down Defrel, but Cesena could not capitalise as Brienza fired over the resulting free-kick.