Premier League strugglers Newcastle flexed their new financial muscle by signing England full-back Kieran Trippier from Atletico Madrid on Friday and he made his debut.
But humble Cambridge were undaunted as Joe Ironside fired in a second-half goal and keeper Dimitar Mitov made a series of saves to seal a famous victory and send their fans into delirium.
Leicester, who beat Chelsea in last season's final to win the competition for the first time, eased through into the draw for the fourth round with goals by Youri Tielemans, James Maddison, Harvey Barnes and Marc Albrighton.
Joao Pedro replied for Watford, managed by Leicester's title-winning manager Claudio Ranieri, but, apart from a floodlight failure that caused a stoppage on the hour, it was relatively smooth progress for Brendan Rodgers's side.
Earlier, Burnley became the first Premier League casualty of the third round when they lost 2-1 to Huddersfield Town who came from a goal down to win late on at Turf Moor through Matty Pearson's 87th-minute header from a corner.
Jay Rodriguez's header had give Burnley the lead but Josh Koroma equalised for the visitors in the 74th minute.
Crystal Palace avoided a similar fate though as they secured a 2-1 comeback victory at London rivals Millwall with goals from Michael Olise and Jean-Philippe Mateta.
Minor league Kidderminster Harriers, the lowest-ranked team to survive until the third round, pulled off a shock when the sixth-tier club beat Championship side Reading 2-1.
Boreham Wood made sure two minor league sides will be in the fourth round when they beat AFC Wimbledon 2-0.
Premier League Brentford enjoyed a comfortable 4-1 victory away to Port Vale thanks to a Bryan Mbeumo hat-trick.