Scott Sinclair ended the hosts' 374-minute wait for a top-flight goal as he coolly slotted home a 14th-minute penalty.
Leroy Lita nodded home his first for the club 10 minutes later before Nathan Dyer secured the points shortly after the break, with Brendan Rodgers' side also keeping a third straight clean sheet at home.
The only concern for the Swans was a nasty head injury suffered by full-back Neil Taylor in an aerial clash with Peter Odemwingie which saw the striker booked.
The visitors had few clear chances, with Shane Long and Graham Dorrans having their best efforts saved by Michel Vorm.
Swansea were boosted by the return of captain Garry Monk, making his first appearance of the season after a foot injury.
Monk's availability to partner Ashley Williams in the centre of defence was a timely boost for Rodgers, with Alan Tate and Steven Caulker injured.
Mark Gower was handed a first start of the season, while Lita began in attack, with Danny Graham out due to a back problem.
West Brom boss Roy Hodgson was without the suspended Gabriel Tamas so Craig Dawson came in at centre-half, with Dorrans making way for James Morrison on the right of the Baggies midfield. Chris Brunt was on the bench after a late fitness test on a knee problem.
There was a minute's silence prior to kick-off in memory of the four miners who lost their lives at Gleision Colliery, as well as Rodgers' father Malachy, who died a week ago.
Unfortunately the tribute was poorly observed by a section of the away support, forcing referee Martin Atkinson to curtail the silence before the minute had elapsed.
Sinclair had an early chance for the hosts but snatched at his shot after a poor Youssouf Mulumbu clearance, before Ben Foster grabbed an Allen shot at the second attempt.
Swansea's early dominance got a deserved reward in the 14th minute as Paul Scharner's clumsy tackle on Allen led to a penalty that Sinclair slotted to Foster's left for the south Wales club's first Premier League goal.
And the second soon followed as Sinclair's header across goal was turned in by Lita at the far post.
Lita was booked after taking his shirt off to celebrate his goal, and as Swansea sat back to admire their lead they surrendered the initiative to West Brom.
Long forced an athletic save from Vorm before two brave blocks from Ashley Williams denied Mulumbu a sight of goal.
Sinclair remained a menace to Albion full-back Steven Reid and another brilliant run from the winger teed up Allen, but the midfielder's shot was blocked by Dawson.
Hodgson made a double change at the break with Brunt and Dorrans replacing Mulumbu and Jerome Thomas, but it made little difference as Swansea further extended their lead within four minutes of the restart.
Dyer raced onto a Lita flick-on and held off Nicky Shorey before firing between Foster's legs to send the Swans faithful into dreamland.
Brunt thought he had pulled a goal back on the hour but his header from Jonas Olsson's ball into the box was ruled out for offside.
Vorm then needed to be at full stretch to keep out a free-kick from Dorrans.
Odemwingie then went in the book after clattering Taylor in an aerial challenge that left the full-back needing prolonged treatment before departing on a stretcher with a head wound.
There were nine additional minutes added as a result but the hosts saw them out for an historic win.