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The Cobblers needed a win to keep the pressure on the teams above them but have lost form at a bad time having taken only one point from the last three games.
Northampton could have taken an early lead when midfielder Andy Holt was allowed the space to head just over the bar from a Jonathan Hayes cross, but Cheltenham went on to create the better chances in the remainder of the first half.
The breakthrough arrived on 39 minutes when Bird's through ball gave Gillespie the chance to run at the defence down the inside-right channel.
Gillespie cut inside and, seeing a gap at the far post, placed his shot with perfect accuracy beyond Bunn to claim his 12th goal of the season.
Northampton turned the tables in the second half and enjoyed far more of the play.
Adebayo Akinfenwa side-footed wide from 12 yards when he really should have scored and left-back Daniel Jones went close with a free-kick from 25 yards.
A deserved equaliser arrived in the 76th minute when a corner taken by Jones was met on the half-volley by Giles Coke and the ball zipped past Cheltenham goalkeeper Shane Higgs.
The Cobblers now look unlikely contestants for the play off positions, but must now concentrate of the visit of Luton Town on Saturday
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