The former England striker nipped in between Dann and the onrushing goalkeeper Wayne Hennessey to secure the Black Cats' first away win of the league campaign.
Shortly after slotting home the winner, however, 33-year-old Defoe hobbled off with a suspected leg injury.
Alan Pardew's toothless Eagles were unable to build on their fine 2-1 win at Liverpool, slipping to their fourth home defeat in seven league games this term.
The hosts created precious few real scoring chances, the industry and application of Wilfried Zaha and Yannick Bolasie lacking a killer final ball.
That profligacy from the hosts allowed Sam Allardyce's side to steal the points - their first league win on the road since the 2-0 triumph at Everton in May.
Sunderland had the only two tangible chances of a dire first half.
Allardyce's wing-back formation twice allowed Patrick van Aanholt in behind the Palace defence, and the first time the left-sided defender ought to have scored.
Defoe blasted wide from all of 12 yards midway through a half punctuated with dross.
The Black Cats opened the second half the stronger but were thwarted from another fine chance when Yohan Cabaye cleared Sebastian Coates' header off the line.
Substitute Bakary Sako's curling effort veered wide as Palace attempted to raise the tempo, before Damien Delaney pulled off a full-stretch interception to deny Sunderland an opportunity on the break.
Sako's night lasted less than 20 minutes, the former Wolves midfielder trudging off to be replaced by Patrick Bamford as Pardew threw two up front.
The clash had cried out for the home team to alter their formation and double their strike force almost from the off, yet Pardew still waited until after the hour mark to make the switch.
While Zaha and Bolasie huffed and puffed in their effort to deliver a telling final ball, Sunderland continued to offer a reasonable counter-attacking threat.
Allardyce's men again wasted a fine opening, however, when replacement Jeremain Lens tapped a tame effort straight at Wayne Hennessey.
Pardew pitched Marouane Chamakh into the front line in place of the anonymous Connor Wickham, but still the hosts struggled for fluency.
Sunderland finally took advantage via the anticipated direct counter, Dann misreading Billy Jones' hopeful ball to allow Defoe a clean run on goal.
The former Tottenham hitman ghosted in and round Hennessey in one touch, before passing into the empty net.
Hennessey had little choice but to leave his area in a bid to cover Dann's misread, but only succeeded in allowing Defoe to tee up his tap-in.
Palace continued to press even through five minutes' added time, but to no avail, with Cabaye curling a free-kick wide to close out a frustrating night for the hosts.
TWEET OF THE MATCH
Gary Lineker @GaryLineker
Well that was a giant cock-up but decent strikers always expect such errors
Defoe did exactly that.
https://twitter.com/GaryLineker/status/668906833659064320
PLAYER RATINGS
CRYSTAL PALACE
Wayne Hennessey 4
Joel Ward 5
Scott Dann 4
Pape Souare 5
Damien Delaney 6
Yohan Cabaye 6
James McArthur 5
Yannick Bolasie 5
Wilfried Zaha 5
Jason Puncheon 4
Connor Wickham 4
Substitutes:
Bakary Sako 5
Patrick Bamford 5
Marouane Chamakh 5
SUNDERLAND
Costel Pantilimon 6
Younes Kaboul 6
Sebastien Coates 5
John O'Shea 6
Billy Jones 6
Patrick van Aanholt 6
Lee Cattermole 6
Yann M'Vila 5
Sebastian Larsson 5
Jermain Defoe 7
Steven Fletcher 4
Substitutes:
Jeremain Lens 6
Duncan Watmore 7
Danny Graham 6
MAN OF THE MATCH
Jermain Defoe: A woeful night for both sides lit up only by Defoe's opportunist finish that settled the contest.
VIEW FROM THE BENCH
Sunderland boss Sam Allardyce caught everyone by surprise with his 3-5-2 formation, but his side were unable to profit despite several good incisions in the first period
Alan Pardew kept Palace's changes to a minimum but hauled off the ineffective Jason Puncheon at half-time, but Bakary Sako could not fare any better
Pardew finally threw two strikers into the mix just past the hour but Palace remained toothless and paid the price
MOMENT OF THE MATCH
Defoe ghosted in between Wayne Hennessey and Scott Dann to poke into space and then tap into the empty net
On a night severely lacking in quality, this moment of natural instinct deserved to seal the deal
MOAN OF THE MATCH
Only referee Martin Atkinson will know why Jermain Defoe was booked for kicking the ball back towards Scott Dann ahead of a free-kick
Dann attempted to steal a few yards on the halfway line, Defoe took exception and nudged the ball backwards, and was duly booked
A fastidious, pointless decision
WHO'S UP NEXT
Crystal Palace v Newcastle (Premier League, Saturday, November 28)
Sunderland v Stoke (Premier League, Saturday, November 28)
Source : PA
Source: PA