Liam Eddy got us off to the ideal start when scoring on 14 minutes, but not for long though as Alex Faux levelled.
Midfielder Olly Brokenshire scored his first goal for the club in the 36th minute to ensure City went in at half-time with a deserved 2-1 lead.
And it was looking that perhaps Olly’s goal was going to be the winner until 16 minutes from time, when the home side levelled through Robbins.
Eddy pounced for his second goal of the night, with a fine volley 11 minutes from time, getting on the end of a cross from Steve Tully, making his debut for the White Tigers.
3-2 to City and we were looking as if the tie was settled. However, with just a minute of normal time left, Robbins netted his second with a free-kick and the tie went straight into penalties.
Another evening ended in disappointment as Bridgwater came out on top 6-5 after missing their first kick.
Cody Cooke and substitute Neil Slateford had their spot kicks saved for City, while Shane White, Les Afful, Steve Tully, Olly Brokenshire and Jamie Lowry were all successful.
City: Chenoweth; Tully, Kendall, Pugh, White; Lowry; Cooke, Brokenshire, Copp (Slateford 89); Eddy, Sims (Afful 70). Subs (not used); D Evans H Evans, Broad.