Marc-Antoine Fortune put the Baggies ahead, firing in Jonas Olsson's flick from a corner, as West Brom sought a first home league win since November.
But Gylfi Sigurdsson inspired a great comeback as he calmly slotted home Neil Taylor's cross to put Swansea level.
Sigurdsson then set-up Graham with a lovely ball that the striker guided in.
Graham's 10th of the season helped move Swansea back into the top-flight's top 10 and onto 30 points, with 14 games remaining, as the newly-promoted side bid to steer away from a relegation battle.
Swansea had the worst away record this season but they fought for a second away win on successive trips to the West Midlands as Brendan Rodgers' men followed up their triumph at Aston Villa in January by condemning West Brom to a fourth home defeat in a row.
West Brom will now be looking nervously over their shoulder as The Baggies are three places - and six points - above the Premier League drop-zone with in-form Sunderland, Chelsea and Manchester United to play in three of their next four games.
And Baggies boss Roy Hodgson will have been frustrated not to win an early penalty in The Hawthorns snowstorm as Gareth McAuley's close-range header seemed to hit Swans defender Ashley Williams on the arm but referee Jonathan Moss was unmoved.
Sigurdsson's cutting edge would have earned plaudits at West Brom but almost as important was him heading Peter Odemwingie's first-half effort off the line as it kept Swansea level, while Fortune's goal-bound follow-up was blocked.
The flashpoint of a pretty uneventful first-half was an off-the-ball scuffle between Williams and Odemwinge, unseen by official Moss.
Swansea's Wales international Williams seemingly stood on Odemwingie's foot during a heated discussion while West Brom's Nigerian striker retaliated with a swing of his elbow.
The second-half quickly exploded into life with three goals inside five minutes that proved decisive.
Fortune's first top-flight goal in 12 months sent relief around The Hawthorns as the French striker fired past Swans keeper Michel Vorm at an angle after Jonas Olsson had flicked on Graham Dorrans' left-wing corner.
But that relief quickly evaporated as Sigurdsson levelled within 37 seconds of the restart.
The midfielder passed Taylor's teasing low cross past Baggies keeper Ben Foster for his first Swansea goal and third goal in as many games against West Brom.
And it was perhaps not surprising that an Icelandic international was excelling in the freezing conditions as Sigurdsson turned provider with a cross whipped in from the right that Graham managed to steer into the corner of Foster's net with a nudge of his left boot.
Graham was denied a chance to extend Swansea's lead thanks to great covering by Olsson as the West Brom defender got to Taylor's low left-wing cut-back before the Swansea top-scorer.
Nathan Dyer also had great chances to put the Swans victory beyond doubt.
But Odemwingie should have equalised late on for West Brom as he missed the target with the goal at his mercy after a lovely flighted ball from Dorrans.
And Swansea clung onto their lead despite another superb chance for Fortune but his injury time shot squirmed under Vorm's body and wide.
LINEUP, BOOKINGS (3) & SUBSTITUTIONS (4)
West Bromwich Albion
01 Foster
03 Olsson
12 Reid
20 Shorey Booked
23 McAuley
07 Morrison Booked
14 Thomas (Tchoyi - 72' )
17 Dorrans (Cox - 71' )
21 Mulumbu (Scharner - 64' )
24 Odemwingie
32 Fortune
Substitutes
13 Fulop
06 Ridgewell
28 Jones
05 Tchoyi
08 Andrews
33 Scharner
31 Cox
Swansea City
01 Vorm
02 Williams
03 Taylor
04 Caulker
22 Rangel
07 Britton Booked
24 Allen
42 Sigurdsson
10 Graham
11 Sinclair
12 Dyer (Routledge - 88' )
Substitutes
25 Tremmel
16 Monk
15 Routledge
17 McEachran
26 Agustien
18 Lita
19 Moore
Ref: Moss
Att: 24,274
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