Edwards opened the scoring in just the second minute while Vine's near-post finish on the hour wrapped up the points for Mike Newell's men.
Palace did come close on a number of occasions most notably when Leon Cort saw a header cannon back off the crossbar in the 80th minute.
They finally got on the scoresheet in the fourth minute of stoppage time when James Scowcroft headed home a Tom Soares cross, but by then the game was over.
Hatters handed a full debut to new £340,000 signing Sam Parkin who started in place of Warren Feeney, Luton's only change to their starting line-up from their last outing, the 1-0 defeat at Wolves two weeks ago.
Palace in turn handed a debut to their own new £2.5million signing from Blackburn Shefki Kuqi, who took the place of Dougie Freedman.
Town kicked off in bright sunshine and took just 67 seconds to open the scoring when Parkin flicked on Sol Davis' long free-kick allowing Vine to nod the ball into the path of Edwards, who ran in behind a static Eagles defence and volleyed past Gabor Kiraly, right-footed from ten yards.
Luton almost added another three minutes later when Parkin released Davis down the left flank with a super first time pass and then just failed to get on the end of the full-back's cross as he raced into the penalty area.
Palace settled and began to dominate the possession, but without creating any meaningful chances and their first shot didn't come until the half hour mark when Mark Kennedy blazed way over the crossbar from 20 yards after Markus Heikkinen had headed out a deep Danny Granville free-kick.
Two minutes later Kennedy came closer when Soares all too easily beat Richard Langley down the left flank and crossed for the former Wolves man to hit a 15-yard effort at Marlon Beresford.
The Eagles' ex-West Ham man skipper Carl Fletcher limped out of proceedings in the 38th minute.
And six minutes later his replacement Michael Hughes came close to grabbing an equaliser when his shot from the edge of the area was cleared off the line by Davis after a Kennedy corner.
Another corner routine five minutes after the break again saw Palace go close when centre-half Mark Hudson headed just over from eight yards after Kennedy's flag kick.
Instead it was Luton who stepped up the tempo and after a short period of pressure went 2-0 up on the hour when Vine latched on to Steve Robinson's inch perfect curling pass from midfield and fired past Kiraly at his near post from the right-hand corner of the six yard box.
Palace almost replied immediately as Davis charged down Soares' shot before Jobi McAnuff seized upon the loose ball and drove just wide from 20 yards.
Palace came again and after Heikkinen was a touch fortunate to get away with some tugging on substitute Clinton Morrison's shirt in the penalty area, fellow Eagles replacement Freedman then swiped a shot over the crossbar from eight yards after Hudson had flicked on another Kennedy corner.
As the game opened up, Luton were next to threaten in the 69th minute when Leon Barnett nodded on a Kevin Foley free-kick for Lewis Emanuel to hit a stinging 14-yard volley that Kiraly smartly turned around the post.
Edwards hit a stinging 20-yard drive inches over the bar five minutes later before Leon Cort crashed a header off the crossbar for Palace with ten minutes remaining after Kennedy's free-kick on the edge of the penalty area.
Town were denied a clean sheet when Scowcroft headed in from six yards following a right-wing cross from Soares.