Bodde hammered home from close range to cancel out Joey Gudjonsson's opener for the Clarets.
The Dutch midfielder collected Angel Rangel's cross for this third goal of the season to deny the visitors, after substitute Gudjonsson looked to have won it with the 59th minute header.
Swans boss Roberto Martinez opted to shake up his side after making four changes and the reshuffle appeared to pay off after a strong start from the home side.
Shaun MacDonald's raid down the right almost brought an opening goal before Bodde's long-range strike was deflected to safety.
But Burnley were inches from taking a 20th minute lead only for Martin Paterson to get a weak connection to strike partner Steve Thompson's knockdown.
The Northern Ireland striker did force Swansea goalkeeper Dorus De Vries into action moments later when his low shot was kept out at the near post.
An error strewn match where chances were few and far between came to life in the second half when Burnley finally strung a move together which culminated in the opening goal from substitute Gudjonsson.
Wade Elliott provided the deep cross and the Icelandic midfielder nodded in at the back post from close range in the 59th minute.
Thompson came close to doubling the visitors' lead moments later when Elliott's dangerous run and cross saw the Scottish striker head straight at de Vries.
Swansea threw on star striker Jason Scotland from the bench and his low shot was pushed around the post by Brian Jensen.
Gorka Pintado fired wide and Bodde saw a header hit the crossbar as the Swans were running out of ideas. But Swansea continued to plug away and were rewarded when Bodde netted the equaliser late on.