Whittingham failed from the spot in the 69th minute and the Eagles took full advantage when Dikgacoi collected a pass from Pablo Counago and lashed home from 12 yards in the 82nd minute.
The result was a further blow to Dave Jones' men after they had lost 2-0 at home to Ipswich on Saturday but it was a massive boot to the Eagles survival prospects.
Palace, unbeaten at home in 10 games, were largely dominant during the opening period and constantly forced the Bluebirds onto the back foot.
In the ninth minute, Palace threatened when David Wright crossed dangerously and James Vaughan headed goalwards from 12 yards.
Bluebirds goalkeeper Stephen Bywater made a comfortable save and two minutes later Palace went close again when Vaughan fired goalwards from six yards and Bywater made a smart stop to turn the ball around a post.
Eagles frontman Vaughan went close again in the 15th minute when Dikgacoi flighted in a teasing cross but Vaughan could only send his header wastefully wide from 12 yards.
Cardiff eventually provided a riposte in the 28th minute when Jay Emmanuel-Thomas hit a 25-yard shot which Julian Speroni was able to save comfortably.
But the momentum remained largely with the home side and they fashioned another decent opening in the 33rd minute.
Neil Danns was afforded the time and space to strike a 12-yard shot wide from 12 yards as Cardiff were again let off the hook.
The visitors, despite fielding three strikers in Bothroyd, Michael Chopra and Craig Bellamy, struggled to make an impression on the visitors' goal.
In the 40th minute Bothroyd hit a 25-yard shot which Anthony Gardner did superbly well to block.
Shortly after the restart, Emmanuel-Thomas also hit a shot from distance which Darren Ambrose did well to block as Cardiff began to up the tempo after their insipid first-half showing.
Bellamy also came more into the game but Gardner blocked the Welshman's cross as he tried to deliver a dangerous ball into the box.
Shortly before the hour mark, Emmanuel-Thomas did superbly well to hit a shot which Speroni tipped around a post.
Cardiff's Kevin McNaughton then drilled a 25-yard shot just wide of Speroni's left-hand post and moments later they went closer still when midfielder Stephen McPhail fired goalwards from long range but Speroni did well to tip the ball over the crossbar.
Cardiff were now dominating but an inswinging corner from Bellamy was headed wastefully over the bar by Dekel Keinan.
In the 69th minute, David Wright handled the ball and Whittingham missed the target before Palace struck the decisive blow eight minutes from time.
Counago did well to find Dikgacoi and he fired home a right-footed shot into the corner from 12 yards.