The Brickies looked set to strengthen their grip on fourth place when Ryan Golding put them ahead with a 19th-minute penalty, his 14th league goal of the season.
But Eastbourne, 20th in the table, replied seven minutes before half-time through Richard Greenfield and then took the lead with a goal from top scorer Danny Curd in the 73rd minute. Although they had to settle for a point in the end, it was enough to lift them two places.
The Ryman North game between Ilford and Redbridge was postponed because of a frozen pitch.
Keep up-to-date with our exclusive email newsletters.
Subscribe