Jay Rodriguez's 15th goal of the season put Burnley ahead but Bellamy struck at the end to ensure the Bluebirds finished level on points with the Swans.
Yet Swansea thrashed relegated Sheffield United at the Liberty Stadium and their superior goal difference meant Dave Jones' men - beaten in last season's Championship play-off final - now face in-form Reading over two legs in the play-off semi-finals.
Rodriguez's strike marked another feather in his cap after he was this week included in the provisional England squad for the European Under-21 Championship finals in June.
The Burnley-born frontman has been named by England coach Stuart Pearce in an initial 40-man party for the tournament in Denmark.
Rodriguez's inclusion follows an initial call-up in February when he won his first international cap as a substitute in a friendly defeat in Italy.
The 21-year-old has enjoyed a fine season and he proved pivotal again against the Bluebirds as Burnley made all the early running.
They could have led inside the eighth minute when winger Ross Wallace engineered the space to hit a left-foot shot which Cardiff goalkeeper Stephen Bywater tipped round a post.
Four minutes later, Cardiff threatened briefly when midfielder Jason Koumas hit a left-footed shot from 12 yards which drew a comfortable save from Burnley stopper Brian Jensen.
The pendulum swung back in Burnley's favour moments later as they opened the scoring in the 13th minute.
Cameroon international defender Andre Bikey produced a superb pass into Rodriguez's path down the right channel.
The striker did the rest with a fine right-footed strike into the bottom left-hand corner from 18 yards.
Rodriguez could have doubled his tally in the 21st minute but saw another fierce shot from distance flash narrowly wide.
In the 29th minute, Koumas had a shot blocked by Jensen and Cardiff then lost Mark Hudson to injury and he was replaced by Darcy Blake.
Cardiff began to build some momentum as the interval approached and in the 35th minute Jay Bothroyd had a shot from 12 yards caught by Jensen.
Bellamy then fired over from 25 yards and Burnley lost Bikey to injury at half-time and he was replaced by Brian Easton.
Shortly after the restart, Bothroyd curled a free-kick from 25 yards which Jensen did well to tip over the crossbar.
In the 51st minute, Cardiff midfielder Chris Burke fired over the crossbar from Peter Whittingham's right-wing corner.
In the 56th minute, Burke was replaced by Jay Emmanuel-Thomas as Jones tried to pep up his attack.
Shortly before the hour, Whittingham's inswinging corner was greeted by Paul Quinn's header but he missed the target from six yards out.
Burnley winger Chris Eagles then went close with a shot that Dekel Keinan blocked and then Bothroyd fired over from 25 yards before Keinan headed over.
In the last minute, Stephen McPhail found Bellamy and he fired a low left-footed shot into the net from 18 yards.