Watford striker Graham struck his 24th goal of the season in the 61st minute after his side had spent much of the game stoutly defending an early advantage given to them by Troy Deeney, while Don Cowie slammed home a third for the visitors in stoppage time.
Victory lifted the Hornets up into seventh place, within four points of the top six, while Ipswich, who have now failed to beat Watford in their last 13 attempts, slipped to a fifth defeat since Paul Jewell replaced Roy Keane as manager.
Ipswich defender Gareth McAuley hit the post with a first-half header as the home side looked to cancel out Deeney's third-minute opener and continued to dominate until Graham's hammer blow, from which the Tractor Boys failed to recover.
Both sides made just one change from the weekend, with midfielder Mark Kennedy returning for Ipswich and Adam Thompson replacing injured Watford defender Lee Hodson.
Watford owed their early lead to Graham, whose cushioned header from the left side of the six-yard box fell for Deeney to sweep the ball home into the bottom corner from 12 yards.
Ipswich's response was swift as Tamas Priskin headed over the crossbar and teenage striker Connor Wickham rifled a 30-yard shot wide.
Watford midfielder Ross Jenkins then blocked Carlos Edwards' shot and Town's McAuley headed Jimmy Bullard's corner over the bar.
McAuley powered another header against the post and Wickham's angled effort from inside the penalty area was well held by Watford goalkeeper Scott Loach.
Hornets midfielder John Eustace failed to trouble Ipswich goalkeeper Marton Fulop with a 25-yard shot before Priskin and then Bullard fired long-range efforts wide as the home side continued to dominate up until half-time as they went in search of an equaliser.
But Watford served warning they were capable of adding to their lead five minutes after break when Thompson raced on to Eustace's cut back and fired an angled drive just over.
Jewell sent on recent loan signing Kieron Dyer in the 53rd minute, the former Portman Road crowd favourite making his first home appearance for the Tractor Boys since returning to the club from West Ham.
But six minutes later, against the run of play, Ipswich fell further behind.
Eustace again caused the damage down the left, picking out Graham with an inch-perfect ball into the penalty area, which was despatched by the former Middlesbrough forward into the bottom corner.
Watford's second goal took the wind from Ipswich's sails and save for Wickham's weak shot from inside the area and Andy Drury's header from substitute Mark Kennedy's cross, the home side had nothing else to offer.
And to complete their misery, Watford midfielder Cowie lashed home a third goal from 25 yards for the visitors in stoppage time.
Ipswich 0-3 Watford: Match Report - view commentary, squad, and statictics of the game as it happened.