West Brom could have been level soon after when Taylor's shocking back-pass went straight to Shane Long, but the striker snatched at the unexpected chance and ballooned it over the bar.
After waiting so long for a goal, Swansea had obviously discovered the knack and conjured a second from a corner.
Sinclair pulled away from his marker and flicked the ball to the back-post for Lita to head home his first Swansea goal, although the striker earned a booking for his shirtless celebrations.
DID YOU KNOW?
The last time Swansea City scored a goal in the top flight was 28 years ago, when Bob Latchford struck in their 2-1 defeat at Manchester United on 7 May 1983
Lita was making his first start of the season after record signing Danny Graham's late withdrawal with a back problem, but has now staked a significant claim to the lone striker's role that Rodgers favours.
Long nearly made amends for his earlier miss with a fierce rising shot, but Swansea keeper Michael Vorm athletically tipped the ball away.
The home side survived one more moment of alarm before half-time when the ball struck Ashley Williams' arm as the Swansea centre-back slid in to block a cross, but referee Atkinson waved play on.
We shot ourselves in the foot - Hodgson
Baggies boss Roy Hodgson replaced Jerome Thomas and Youssouf Mulumbu with Chris Brunt and Graham Dorrans for the second half.
But the changes made little difference as the visitors soon found themselves three behind.
Lita climbed above his marker to flick Vorm's long punt into the path of Dyer, who took the ball in his stride before shooting through Foster's legs.
West Brom at last managed to exert some extended pressure and Brunt thought he had pulled a goal back on the hour, but his header from Jonas Olsson's cross was ruled out for offside.
It then took a sensational save from Vorm to deny Dorrans, the Dutchman somehow getting a hand to a free-kick seemingly destined for the top corner.
Swansea's day was marred, however, when Odemwingie clattered into the back of Taylor as he chased a high ball.
The clash saw the stretcher, oxygen and a neck brace come out for Taylor, while the West Brom striker got a yellow card for his part.
There was a nine-minute delay while medical staff treated the stricken Wales left-back - who thankfully regained consciousness in the dressing room - who was replaced by Federico Bessone.
When play resumed Long almost grabbed a consolation for West Brom, only to again find Vorm in excellent form.
But Swansea finished the game on top, as they had been for most of the match, with midfielder Allen almost grabbing a late goal to cap his excellent performance.
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