The League Two side frustrated the Premier League champions throughout a goalless first half at Rodney Parade and came agonisingly close to taking the lead despite City dominating possession on an uneven surface they initially struggled to adjust to.
Newport's admirable resistance was broken when Leroy Sane fired home from close range early in the second half, the winger scoring his 12th goal of the season on an afternoon when his customary flair was in short supply.
Foden then scored a brace either side of an 88th-minute Padraig Amond goal that gave Newport brief hope and extended his record of netting in every round of the competition.
Riyad Mahrez added a fourth in second-half stoppage time to secure a scoreline that somewhat flattered City, who survived some nervous moments in South Wales but will fancy their chances of going all the way after avoiding a potential banana skin.
City's hearts were in their mouths after 14 minutes. A long Newport throw was flicked on by Jamille Matt and Tyreeq Bakinson applied a far-post header but Ederson flung himself across his goal and parried the ball away.
Sane and David Silva were both denied shooting opportunities in the Newport box as the visitors struggled to find their best rhythm on a bobbly surface.
Sane did hit the crossbar with a rising shot from close range after a neat one-two with Silva and City were denied a penalty when another Sane shot hit County captain Mark O'Brien's arm towards the end of a goalless first half.
The second half was six minutes old when Gabriel Jesus played Sane in and the Germany international struck a left-footed shot from the corner of the six-yard box that hit goalkeeper Joe Day in the face on its way into the Newport net.
Day should have done better with Foden's shot from the edge of the box after 75 minutes but nothing should be taken away from the City teenager, who carried the ball from just inside his own half before putting the game beyond Newport with a low finish.
City substitute Aymeric Laporte gifted Amond the chance to poke the ball over Ederson and into the net with two minutes left, but Foden responded by dribbling past three Newport defenders and rifling into the net from just inside the box.
Mahrez then latched onto a cute through-ball from John Stones before lashing City's fourth past Day - another goal the keeper will want to avoid seeing again after he was beaten close to his body.
What does it mean? Silverware beckons for City
The Premier League title race is hotting up and Chelsea await City in the League Cup final but this victory means Pep Guardiola's men still have a chance of glory in four competitions.
Jesus adjusts to the conditions
Jesus waited 51 minutes before making a positive contribution to his side's attacking efforts but then provided two assists in a single game for the first time in a City shirt.
Bad Day at the office
Day got a touch on two of City's goals but was beaten for power and he will not enjoy watching replays of Newport's biggest game of the season.
Key Opta Facts
5 - Padraig Amond is the first player to score in each of the first five rounds in a single FA Cup campaign (replays included) since Jon Stead for Bradford City in 2014-15. Habit. pic.twitter.com/CBdHgRQNNA— OptaJoe (@OptaJoe) February 16, 2019
- Newport have lost just two of their last 12 FA Cup matches (W7 D3), with each defeat coming against one of the Premier League's 'big six' sides (they lost to Tottenham in 17-18).
- Sane has had a hand in 25 goals in 32 appearances in all competitions this season (12 goals, 13 assists), including registering nine goal involvements in his last eight.
- Jesus has been directly involved in 21 goals in 34 games this season (16 goals, 5 assists), one more than he managed in 42 games last season (17 goals, 3 assists).
- All five of Foden's City goals have come in domestic cup competitions (3 in the FA Cup, 2 in the League Cup).
- Amond has scored five goals in this season’s FA Cup, more than any other player.