Pep Guardiola's side had failed to score in their previous five visits to Spurs' home and that unwanted record looked set to extend until Ake bundled home from close range after goalkeeper Guglielmo Vicario flapped at Kevin De Bruyne's corner.
All eyes had been on De Bruyne and Tottenham playmaker James Maddison after a lacklustre first half and while the former helped break the deadlock, it was an uncharacteristic City goal that fired them to a seventh straight win.
Ange Postecoglou's team could have little complaints and this cup loss means the club's trophy drought will almost certainly extend into a 17th year.
City's notoriously poor record at Tottenham Hotspur Stadium was well documented and Guardiola's side almost broke their duck in the fifth minute when Oscar Bobb tapped home on the goal line.
Bobb pounced after Vicario had denied Phil Foden from close range, but City youngster Bobb was adjudged by VAR to be marginally offside.
The visitors continued to seek a maiden goal at this venue and after Bernardo Silva and Mateo Kovacic had efforts blocked, Guardiola let out his first sign of exasperation when Josko Gvardiol sent a cross into the stand.
Spurs had barely had a kick but were given some unexpected respite when a three-minute delay to proceedings occurred due to one of Paul Tierney's assistant referees having an issue with his technical equipment.
When play did get back under way, normal service resumed and a succession of last-ditch defending denied City in the 41st-minute with Bobb's left-footed strike thwarted by Pedro Porro's block.
Micky van de Ven made a vital intervention straight after half-time when Julian Alvarez had a sight of goal before Tottenham finally caused a moment of panic for the away side.
It reenergised the home crowd although Guardiola turned to his ace in the pack with 65 minutes on the clock when De Bruyne was introduced.
Postecoglou followed suit eight minutes later and Maddison earned a hero's reception on his return.
But Ake had other ideas and when Vicario blundered the City defender was on hand to poke home and send the holders through to the last 16.
Postecoglou cut a frustrated figure after the loss.
"We fought hard to stay in the game but to be honest that's all we did, we stayed in the game.
"They're a top team. They're the benchmark. We're not there yet and we're under no illusion about that.
"I just felt that all of the second half was ok, the first half we were just too passive in a lot of our play and allowed them to get a rhythm."
Mauricio Pochettino's team had scored six in their previous outing, dispatching Middlesbrough to reach next month's Carabao Cup final, but found Unai Emery's Premier League high-flyers a more obdurate hurdle as a combined 23 shots from both sides failed to yield a winner.
It was a theme of the night in west London as a replay at Villa Park beckons.
In other matches, Bristol City drew 0-0 at home to Premier League club Nottingham Forest and will have to travel to the City Ground while Sheffield Wednesday will have to replay their tie with Coventry City at the Building Society Arena after their 1-1 result at Hillsborough Stadium.