This is a loosely based discussion. So take it for what it's worth. Here are the links I used for the info.
http://www.stevetheump.com/Payrolls.htm
http://www.baseball-reference.com/postseason/
Last 10 seasons starting in 2002. World Series Champ in bold. First 2 teams listed in each league played for league championship, with winner listed first.
2002
AL
Anaheim Angels ($62M - 15th)
Minnesota Twins ($40M - 27th)
Oakland A's ($39M - 28th)
NY Yankees ($126M - 1st)
NL
Giants ($78M - 10th)
Cardinals ($74M - 13th)
Braves ($93M - 7th)
Diamondbacks ($102M - 4th)
Missed postseason: Red Sox (2nd in payroll), Rangers (3rd), Dodgers (5), Mets (6), Mariners (8), Indians (9)
2003
AL
Yankees ($152M - 1st)
Red Sox ($100M - 6th)
Twins ($55M - 18th)
A's ($50M - 23rd)
NL
Marlins ($49M - 25th)
Cubs ($80M - 11th)
Giants ($83M - 9th)
Braves ($106M - 3rd)
Missed postseason: Mets (2), Dodgers (4), Rangers (5), Mariners (7), Cardinals (8)
2004
AL
Red Sox ($125M - 2nd)
Yankees ($182M - 1st)
Angels ($101M - 3rd)
Twins ($53M - 19th)
NL
Cardinals ($75M - 11th)
Astros ($74M - 12th)
Dodgers ($89M - 7th)
Braves ($88M - 8th)
Missed postseason: Mets (4), Phillies (5), Cubs (6), Giants (9)
2005
AL
White Sox ($75M - 13th)
Angels ($95M - 5th)
Red Sox ($121M - 2nd)
Yankees ($206M - 1st)
NL
Astros ($76M - 12th)
Cardinals ($93M - 6th)
Padres ($63M - 17th)
Braves ($85M - 10th)
Missed postseason: Mets (3), Phillies (4), Giants (7), Mariners (8), Cubs (9)
2006
AL
Tigers ($82M - 14th)
A's ($62M - 21st)
Yankees ($195M - 1st)
Twins ($63M - 19th)
NL
Cardinals ($89M - 11th)
Mets ($101M - 5th)
Dodgers ($98M - 6th)
Padres ($70M - 17th)
Missed postseason: Red Sox (2), Angels (3), White Sox (4), Cubs (7), Astros (8), Braves (9)
2007
AL
Red Sox ($143M - 2nd)
Indians ($62M - 23rd)
Yankees ($189M - 1st)
Angels ($109M - 4th)
NL
Rockies ($54M - 25th)
Diamondbacks ($52M - 26th)
Cubs ($99M - 8th)
Phillies ($89M - 13th)
Missed postseason: Mets (3), White Sox (5), Dodgers (6), Mariners (7), Tigers (9)
2008
AL
Rays ($44M - 29th)
Red Sox ($133M - 4th)
White Sox ($121M - 5th)
Angels ($119M - 6th)
NL
Phillies ($98M - 12th)
Dodgers ($118.5M - 7th)
Cubs ($118M - 8th)
Brewers ($80M - 15th)
Missed postseason: Yankees (1), Mets (2), Tigers (3), Mariners (9)
2009
AL
Yankees ($201M - 1st)
Angels ($114M - 6th)
Red Sox ($122M - 4th)
Twins ($65M - 24th)
NL
Phillies ($113M - 7th)
Dodgers ($100M - 9th)
Cardinals ($88M - 13th)
Rockies ($75M - 18th)
Missed postseason: Mets (2), Cubs (3), Tigers (5), Astros (8)
2010
AL
Rangers ($55M - 27th)
Yankees ($206M - 1st)
Rays ($72M - 21st)
Twins ($97M - 11th)
NL
Giants ($98M - 10th)
Phillies ($141M - 4th)
Reds ($72M - 19th)
Braves ($84M - 15th)
Missed postseason: Red Sox (2), Cubs (3), Mets (5), Tigers (6), White Sox (7), Angels (8), Mariners (9)
2011
AL
Rangers ($92M - 13th)
Tigers ($105.7M - 10th)
Yankees ($202M - 1st)
Rays ($42M - 29th)
NL
Cardinals ($105M - 11th)
Brewers ($85M - 17th)
Phillies ($173M - 2nd)
Diamondbacks ($53M - 25th)
Missed postseason: Red Sox (3), Angels (4), White Sox (5), Cubs (6), Mets (7), Giants (8), Twins (9)