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But Cheltenham forward Damien Spencer scored a ten-minute hat-trick to give the Robins a 3-1 lead.
He rounded Paul Musselwhite to tap in the equaliser in the 31st minute after breaking from midfield unchallenged.
Spencer was lucky to stay on the pitch two minutes later when he tussled with Richard Hinds but escaped with a yellow card, while Town had another scare a minute later. Damien Delaney had a shot parried out for a corner, and then a goalmouth scramble was eventually cleared.
City were then punished as Spencer scored twice in 90 seconds to complete his hat-trick. Elliott was beaten on the left and when the ball was sent in, Fyfe had the better of Thelwell to play it across goal where Spencer finished.
The visitors went 3-1 up when a terrible clearance by Musselwhite was headed back in by Yates, and an unmarked Spencer beat the City keeper to it to score.
But Jason Price gave City a lifeline before half-time after tapping in Allsopp's header in the 42nd minute.
Peter Taylor sent his side out for the second half looking for an equaliser, and Allsopp should have scored it after controlling well, but he hit the bar with the shot.
However, he made amends a minute later when Price passed from the right and Allsopp's shot found the net to make it 3-3 after 51 minutes.
Ben Burgess put a shot wide, Odejayi cut out a pass to deny Allsopp going clean through and Price was denied a penalty as Cheltenham clung on for their point. Price almost won it for City late on but Fyfe cleared off the line after the winger headed Keates' corner at goal.
Cheltenham were most threatening in the final minutes, and Musselwhite palmed McCann's free kick over after Keates fouled needlessly 25 yards out.
Uncertain defending by Musselwhite and his defender Marc Joseph - who appeared to be limping - also put City under pressure late on, but it ended 3-3 and City's unbeaten home record is maintained.
City line-up (4-4-2): Musselwhite; Thewell, Hinds (Whittle 72), Joseph, Delaney; Price, Green (Keates 72), Ashbee, Elliott (Forrester 82); Burgess, Allsopp.
Un-used subs: Holt, Melton.