The Robins almost snatched the lead on 10 minutes through Neil Kilkenny, who hit a powerful right-footed shot against the bar.
The Tangerines had a good chance early in the second period through Ince, whose curling strike was saved by Heaton.
Heaton was called into action again on the hour mark, this time to deny Phillips, as the visitors began to take the game to Bristol.
Nathan Delfouneso was the next to test the shot-stopper, but it wasn't all one-way traffic. Jonathan Stead forced a save out of Matthew Gilks at the other end on 78 minutes.
Bristol City were awarded a penalty against the run of play on 79 minutes when Gilks brought down Albert Adomah inside the area. Steve Davies stepped up and powered home the spot kick to make it 1-0 to the Robins.
The hosts were on course for a rare win, but there was late heartache for them when Blackpool won a penalty of their own in the final minute.