In a turgid game of few clear chances 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 sights of goal.
Jussi Jaaskelainen made two point-blank stops to deny Chico Flores and Wilfried Bony at the other end, but the hosts were most aggrieved referee Phil Dowd's decision to turn down a stoppage-time appeal for handball against Joe Cole.
The West Ham substitute had his arm raised as he leapt to block a Nathan Dyer cross but the game ended goalless.
Allardyce, who did not attend his post-match press conference, again opted to start with a side containing no recognised striker, and their 4-6-0 formation stymied Swansea's passing game, while they carried a potent threat from set-pieces at the other end.
He told the BBC: "We should have had it over and done with by half-time.
"We showed our classic counter-attacking football, but unfortunately finishing qualities let us down.
"We limited Swansea to one header from a set-piece. In the second half we didn't pass it as well as in the first. It's disappointing.
"It's our fifth clean sheet in nine matches but we have only nine points in total and that's not good enough.
"If we scored in some of these matches we'd be sitting pretty. They were simple chances and not difficult ones."