Chelsea loan striker Bamford blasted straight at a grateful Lukasz Fabianski, spurning Palace's finest chance to secure victory at Selhurst Park.
Swansea held firm for a draw that could prove vital in their continued Premier League survival fight, a second clean sheet after their 1-0 Boxing Day win over West Brom.
Alan Pardew may well scout around to add a striker to his Palace squad in January, given his side's shortcomings up front in the absence of the injured Conor Wickham.
The ineffectual Marouane Chamakh was hauled off after a first half that elicited no chances for either side
Bamford at least worked several openings, but was left to rue the miss of the day.
Yohan Cabaye will also now miss Palace's home clash against Chelsea on Sunday after collecting his fifth league booking.
Palace again struggled without Yannick Bolasie, out for some 10 weeks with hip trouble, with Wilfried Zaha shouldering a heavy workload out wide.
No one at Selhurst Park will linger long over a dire first half, neither side able to muster a single goal-scoring chance.
Cabaye flashed wildly high and wide twice, while Zaha turned down a fine opening for a shot from 16 yards.
Swansea were just as abject and profligate in possession as the hosts, Jonjo Shelvey's stray pass handing Zaha the gift he then inexplicably ignored.
Former Liverpool man Shelvey almost made immediate amends, threading a smart ball through for Marvin Emnes, only for Wayne Hennessey to rush out and claim authoritatively.
Palace skipper Mile Jedinak surely has no idea how he lost his bearings to the extent he nodded a free header in the Swansea area away from goal.
And pacey Swansea forward Mo Barrow burst past the lumbering Brede Hangeland, only to have the ball lifted off his toe by Scott Dann just when expecting to unleash an effort on goal.
Palace boss Pardew hauled off Chamakh in favour of Bamford at the break
The hosts finally carved out a chance when play resumed, but no thanks to the personnel switch.
Fabianski denied Palace the opener, palming away Hangeland's header after a well-worked free-kick.
Jason Puncheon almost carved an opening for Bamford with an ambitious attempted through-ball, but still the deadlock remained.
Barrow blasted Swansea's first shot on target from 20 yards, but Hennessey met his rare task with relative comfort.
Bamford then blasted straight at Fabianski as Palace battled again for the ascendancy
The 22-year-old simply had to score with a strike from the penalty spot.
Ashley Williams denied Palace again with a smart block from Pape Souare's cutback as Swansea continued to frustrate the visitors.
Zaha turned Kyle Naughton inside-out, only to see his curling effort spin just wide of the statuesque Fabianski.
Hangeland upended Bafetimbi Gomis in another crude challenge in shooting range, allowing Gylfi Sigurdsson to take aim, but Hennessey was able to tip his accurate effort around his left post.
Tweet of the Match
Back to back 0-0, 5 games unbeaten and 5th in the @premierleague, I mean, would have taken that all day long in August
@CPFC tough 3 days
- Ex-Palace striker @Mark__Bright takes the positives after a grim game.
RATINGS:
PALACE:
Wayne Hennessey 7 (Out of 10)
Joel Ward 6
Brede Hangeland 4
Scott Dann 6
Pape Souare 8
Mile Jedinak 6
James McArthur 6
Yohan Cabaye 6
Jason Puncheon 6
Wilfried Zaha 7
Marouane Chamakh 5
Replacements:
Patrick Bamford 5
Chung-yong Lee 6
Jordon Mutch 6
SWANSEA:
Lukasz Fabianski 7
Kyle Naughton 6
Ashley Williams 8
Jordi Amat 6
Neil Taylor 6
Jack Cork 6
Jonjo Shelvey 6
Matthew Grimes 6
Marvin Emnes 7
Bafetimbi Gomis 7
Mo Barrow 6
Replacements:
Ki Sung Yeung 6
Gylfi Sigurdsson 7
Kyle Bartley 6
STAR MAN:
Ashley Williams: The Swansea captain epitomised his side's battling qualities, putting in several vital interventions to ensure the visitors sneaked away from south London with a crucial point
MOMENT OF THE MATCH:
Williams' one miscue of the match handed Patrick Bamford the chance to claim his first goal on his ninth appearance for Palace, but the on-loan Chelsea man could only draw a routine save from Lukasz Fabianski
VIEW FROM THE BENCH:
Palace boss Alan Pardew will be left frustrated by his side's inability to claim the winning goal
The Eagles manager hauled off Marouane Chamakh at the break, his lone striker struggling to make an impact.
Chamakh's replacement Patrick Bamford had the chance to seal the points but spurned the best opening of the match.
Swansea will have been delighted to edge away with a point that owed everything to their defensive organisation.
MOAN OF THE MATCH:
Referee Neil Swarbrick was every inch as sluggish as both sets of players
But not even holiday season over-indulgence could excuse his ponderous approach.
After labouring over decision after decision, the man in the middle incensed the already-frustrated crowd by delaying a Palace free-kick to mark out the 10 yards with his magic spray.
Some 30 painful seconds later, the end result was that everyone was already in the right place - obvious to all but the man in charge.
WHO'S UP NEXT:
Crystal Palace v Chelsea (Barclays Premier League, Sunday January 3)
Manchester United v Swansea (Barclays Premier League, Saturday January 2).
Source : PA
Source: PA