Watford, beaten Championship play-off finalists last season, went 2-0 up inside 30 minutes through Fernando Forestieri and Troy Deeney to leave free-scoring City at risk of a shock defeat.
However, Aguero struck three times in the last half an hour, with Aleksandar Kolarov also on target, to send last season’s beaten finalists into the fifth round.
With Watford 15th in the Championship, they were given little chance of pulling off an upset at the Etihad Stadium, but they took the lead in the 21st minute when Forestieri beat Costel Pantilimon.
It was a reward for the visitors’ bold start to the game and nine minutes later they made it 2-0, with Deeney finishing off a sweeping move involving Sean Murray and Ikechi Anya.
City’s struggles prompted manager Manuel Pellegrini to send on defenders Vincent Kompany and Pablo Zabaleta for the second half and with half an hour remaining the hosts pulled a goal back through Aguero.
The Argentine then plundered an equaliser in the 79th minute, slipping the ball onto his left foot and curling home his 24th goal of the campaign following a raid led by Jesus Navas.
Watford were still within sight of a money-spinning replay, but with three minutes remaining Kolarov deceived visiting goalkeeper Jonathan Bond from distance to make it 3-2, before Aguero completed his treble at the death.
“I was very worried at half-time because I talked with the players a lot before the match and said that in the FA Cup, teams that play in a lower division have a lot of motivation,” Pellegrini said.
“I didn’t underestimate Watford. I didn’t think it would be easy and I told the team this. The players must not believe they have an easy game, but I feel that they had a very good reaction in the second half.”
Deeney said that Watford could be proud of their performance.
“People said it was going to be 10-0 and nobody gave us a chance, but we knew we are not a bad set of players,” said the striker. “City are one of the biggest teams in the world, but we showed no fear.”
Liverpool also avoided a potential banana skin by winning 2-0 at second-tier Bournemouth.
Beneath bright winter sunshine on the English south coast, Liverpool withstood an enterprising start from their Championship hosts before taking the lead in the 26th minute through Victor Moses’s low drive.
Daniel Sturridge added a second goal with half an hour remaining, running onto Suarez’s dinked pass and tucking the ball past goalkeeper Lee Camp in front of watching England manager Roy Hodgson.
Liverpool manager Brendan Rodgers praised his opposite number Eddie Howe for the way that Bournemouth had approached the game.
“When you have a team that want to play football, they are the most courageous players on the field,” he said.
“Bournemouth are a very good side and I give credit to them first of all. Eddie is similar to myself in terms of the philosophy of football.”
Premier League Crystal Palace were the day’s biggest casualties, losing 2-1 at holders Wigan Athletic, who were relegated to the Championship after stunning City 1-0 in last season’s final.
Ben Watson, who scored the winning goal at Wembley last May, set Wigan on their way in the 36th minute, with James McClean netting a 78th-minute winner after Aaron Wilbraham drew Palace level.
After stunning Manchester United in the League Cup semi-finals in mid-week, Sunderland toiled to overcome non-league Kidderminster Harriers 1-0, with Charis Mavrias scoring the only goal in the fifth minute.
Swansea City, Cardiff City, Southampton and Hull City also overcame lower-league opposition, while Steven Naismith scored twice as Everton won 4-0 at third-tier Stevenage.