Davies slammed a penalty past Matt Gilks in the 81st minute after Albert Adomah was fouled, ending a goal drought stretching back nearly seven hours, but Ince had the final say on the stroke of 90 minutes.
The home side who had lost seven games in a row, came agonisingly close to scoring for the first time in four games after just nine minutes when Neil Kilkenny drilled the ball against the bar from 35 yards out, while at the other end Ince cut inside but saw his effort well saved by Tom Heaton.
Adomah, back from mid-week international duty with Ghana, was the next winger to cut inside and try his luck but his left-footed shot failed to test Gilks after a clever one-two with Kilkenny.
Nathan Delfouneso and Kevin Phillips both tried their luck before the interval, the latter with a speculative 35-yard effort, but Bristol City's defence held firm.
Gilks needed to be alert to keep the game goalless shortly after the break, the Scot racing off his line to deny Jon Stead after the ball broke to the former Huddersfield striker after an under-hit backpass from Ian Evatt.
Ince then warmed the gloves of Heaton from distance in the 51st minute before Craig Cathcart put his body on the line to bravely block a strike from Marvin Elliott on the hour mark.
The visitors came incredibly close to breaking the deadlock in the 65th minute when Delfouneso ran clear and his effort from 20 yards hit the post. When the ball fell to Ince, the winger had time to pick his spot but Heaton pulled off a top-class save to keep out his effort from 12 yards.
In the 78th minute, Gilks hauled down Adomah after the winger chased a long ball and referee Gavin Ward pointed to the spot.
McInnes sent on Davies for Stead while the penalty was waiting to be taken and their top scorer stepped up to send Gilks the wrong way with his first touch.
But just when it looked like City would celebrate their first victory since September 18 the ball hit debutant Matthew Bates' hand inside the box and Ince stepped up to net the equaliser.