Brendan Rodgers' side have carved out a reputation for their free-flowing football and looked to have claimed an efficient if unspectacular victory courtesy of a goal which would have received critical acclaim were it scored in Tuesday night's Champions League clash at the Nou Camp.
The recalled Stephen Dobbie was the man to get the final touch on a well-worked passing move midway through the first half.
But the hosts lost their urgency after the breakthrough and Watford, capitalising on an attacking change in formation, levelled proceedings in the 78th minute courtesy of Graham's 19th league goal of the season.
Swans will now look back on Saturday's defeat at Scunthorpe and tonight's draw as a real missed opportunity in the promotion race.
In contrast, the Hornets will be buzzing with the point at the Liberty Stadium having ended an eight-match winless streak at the weekend.
Both sides showed two changes, while Watford boss Malky Mackay mirrored City's 4-5-1 formation to match up the midfield.
Nevertheless it was the hosts who enjoyed the first clear-cut chance of the evening courtesy of an incisive pass from Joe Allen.
Yet recipient Nathan Dyer got caught in two minds and saw his eventual attempt turned around a post by Scott Loach, with Alan Tate forcing a solid effort from the resulting corner.
Allen was proving a real livewire in the early exchanges as Swansea typically enjoyed plenty of possession.
Watford were looking to frustrate the home side although Graham found himself in a promising position in the final third only to fire horribly wide.
Dyer forced a parried save from Loach at the other end before Swansea took the lead on the back of some great approach play.
A flowing move found its way to midfielder Mark Gower who laid on an inch-perfect pass to find Dobbie who poked the ball past Loach.
The pace of the game slowed considerably after Swansea's opener, something which suited the Hornets.
Swans stopper Dorus de Vries had to produce a double stop to keep out Will Buckley as Watford weaved attacking openings.
And the Dutchman again saved well from Graham after he got in behind the home defence, although Swansea held out to keep their one-goal advantage at the break.
That almost became 2-0 within minutes of the restart as Loach tipped over Dobbie's 20-yard effort after Watford lost possession in a dangerous area.
Mackay changed things just before the hour mark, introducing Marvin Sordell and switching to a 4-4-2 system.
And the change immediately provided effective as Watford had more men in the attacking third, with Troy Deeney heading straight at De Vries.
Swansea desperately needed a second goal and they were punished 12 minutes from time.
Deeney found Graham with a great ball and the prolific striker finally got the better of De Vries with an exquisite finish.
Both sides enjoyed chances to nick it but, rightly so, the clash ended all square.