Elliott almost set up a second just moments later, but Danny Allsopp somehow missed from just a few yards out after beating Jones to the ball.
Barmby limped off in the 27th minute after injuring his ankle in a tackle, with Jon Walters coming off the bench.
Green should have got another goal but shot straight at Jones, while Allsopp could have done better with chances. Blackpool made City rue those misses by equalising in the 31st minute. A defensive lapse allowed the ball to get to Scott Taylor, who turned and fired a brilliant shot into the net.
City regained their lead in the 57th minute courtesy of an Elliott strike. France got his head to the ball and Walters laid it down to Elliott, whose powerful shot may have taken a slight deflection as it went into Jones's goal.
Blackpool had City on the backfoot in the latter stages of the game as they sought another equaliser, but it wasn't forthcoming. At the other end, Walters and France had chances to wrap up the win after going clear but couldn't find the target so the score finished 2-1 to City.
City line-up (4-4-2): Myhill; Joseph, Cort, Delaney, Dawson; France, Green, Ashbee, Elliott; Allsopp (Price 61), Barmby (Walters 27).
Un-used subs: Duke, Lewis, Keane.