The Hammers' dead-ball threat troubled their hosts all afternoon, with Angel Rangel hooking a long Razvan Rat free-kick off the line while Guy Demel and Kevin Nolan also spurned good chances.
Jussi Jaaskelainen made two point-blank stops to deny Chico Flores and Wilfried Bony at the other end, and the hosts also had a late penalty appeal for handball against Joe Cole turned down as Swansea were left frustrated as the game finished 0-0.
While neither side picked up the three points, Sam Allardyce will be the happier of the two managers as his decision to again play without a recognised striker for much of the game paid dividends as they restricted Swansea's space in the middle of the park.
They also highlighted Swansea's discomfort defending set-pieces, which has been no secret given their relative lack of height, and it proved a recurring theme throughout the contest and will almost certainly have been noted by Cardiff ahead of next weekend's south Wales derby.
During the early going, West Ham were happy to sit deep and soak up pressure and hope to capitalise on counter-attack or set-piece opportunities.
On several occasions they also attempted to get Stewart Downing one-on-one with Neil Taylor, and one such instance saw the England winger cut in and force a sharp save from Michel Vorm, with Demel prodding over from the resulting corner.
Swansea's first chance fell to Alvaro Vazquez, who had a poor afternoon, as his low header was held by Jaaskelainen after a lovely outside-of-the-foot cross from Wayne Routledge.
Both sides then contrived to pass up excellent opportunities in the 20th minute.
Flores headed straight at Jaaskelainen from point-blank range and West Ham collected the loose ball and broke through Ricardo Vaz Te, who was later forced off through injury.
He found Ravel Morrison but the Hammers starlet dragged wide of the far post.
Vorm enjoyed a lucky escape as West Ham's aerial bombardment continued in earnest, the Dutchman came for Rat's long free-kick and missed it under pressure from Nolan, with Rangel hooking the ball clear as it headed for the back of the net.
Vorm then punched a corner straight to Morrison, with several Swansea bodies rushing out to block the shot.
Substitute Matt Jarvis, on for Vaz Te, beat Rangel and picked out Nolan, with the West Ham captain spooning over with Morrison better-placed behind him.
Swansea raised the tempo after the break and Nathan Dyer crafted an opening for Vazquez by spinning away from Winston Reid and Nolan, but the striker could not get a touch on the winger's inviting ball.
But the hosts' set-piece demons would not go away, with Vorm unconvincingly punching a sliced Michu clearance and Jarvis hoist when under pressure, with Nolan flicking over the bar from the latter.
Allardyce threw on Carlton Cole for his second West Ham debut, while Michael Laudrup introduced Bony for the ineffectual Vazquez.
The Ivorian thought he had given Swansea the lead with 19 minutes to play from, of all things, a set-piece.
De Guzman's corner fell to the striker but Jaaskelainen pushed out his shot from the edge of the six-yard box, with Mark Noble clearing Flores' follow-up header off the line.
Jarvis then headed over under pressure from Rangel at the other end after Mohamed Diame had escaped the home defence.
Swansea thought a penalty should have gone their way in stoppage time, but Phil Dowd waved away appeals that Joe Cole had handled a Dyer cross.