10 minutes on and Swansea should have gone in front. A defensive mixup saw Paul Musselwhite leave his goal open, but John Anderson was in the right place to head James Thomas' shot off the line.
They were made to pay as a minute later Green was the provider for Phil Jevons to make it 1-0. The midfielder played an inch-perfect ball from the left wing to Jevons, who beat his man to convert past Freestone and get his first goal in City colours.
Paul Musselwhite was called into action 7 minutes before half time. Dave Theobald headed a corner goalwards but Musselwhite got a hand to it to tip it away.
However Musselwhite could do nothing when Swansea levelled matters 6 minutes into the second half. Uncertainty left Thomas unmarked from a long throw, and he looped a header over Musselwhite to make it 1-1.
City had lost their first half composure, and the closest they came to troubling Swansea were headers from Ian Ashbee and Phil Jevons.
The visitors also had chances to claim three points, with Watkins having the best of them. Williams pulled a ball back from the byline, and Musselwhite saved on the line from Watkins' effort.
City line-up (4-4-2): Musselwhite; Regan, Whittle, Anderson, Smith (Edwards 57); Green, Ashbee, Keates, Williams (Johnson 57); Alexander (Dudfield 57), Jevons.
Un-used subs: Glennon, Bradshaw.