The wing powered over for his 50
th competitive try for the region and completed his brace on 36 minutes as Danny Wilson's side lead 17-10 at the break.
Munster fought back in the second-half and turned the screw thanks to the boot of Ian Keatley and a Dave Kilcoyne try.
But the Blues responded with a stunning team effort finished by Dan Fish to snatch the spoils in stunning fashion.
The encounter got off to a cagey start with Munster opting to put the ball in the air or to play for the corners but the Blues soon displayed their attacking intent.
They continually probed Munster’s defence and created the first scoring opportunity following a fine break from Dan Fish on the counter attack.
The full-back beat several defenders before linking with Nick Williams, who in turn fed James. Possession was maintained and from the next phase the ball was spread to Allen on the left but he was bundled into touch.
It did not take long however for Cardiff Blues to take the lead with James popping up in midfield to take Fish’s pass and ease between two defenders, before stepping inside Andrew Conway to claim his 50
th competitive try for the region.
Anscombe added the extras to give his side a 7-0 lead after just eight minutes but it did not last long as Munster hit back.
The Blues failed to deal with a kick through and following a scrappy exchange an offload set Conway clear to crash over.
Ian Keatley was on target to draw level but Anscombe soon nudged Cardiff Blues in front with a difficult penalty on the right.
Munster were putting plenty of pressure on Wilson’s men with their sheer intensity and intelligent kicking game but his defence held firm with Josh Navidi claiming one standout turnover.
Anscombe was narrowly off target with a long-range penalty on 22 minutes and on the half hour mark Keatley levelled the scores once more.
On 32 minutes Gethin Jenkins turned down a kick at goal and Anscombe dinked the ball into the corner.
Navidi won the ball and they launched a driving lineout. The Blues were held at bay but hammered away at the Munster line before Ellis Jenkins twisted over.
Referee Ian Davies appeared to award a try before opting for the TMO and the score was ruled out for obstruction.
The Blues however were not to be deterred by the decision and they struck from distance on 36 minutes.
Rey Lee-Lo created the opening after receiving a beautifully timed pass from Anscombe, he skipped past Dan Goggin found James in support, who displayed his pace and power to claim his second.
Anscombe converted the score to make it 17-10 and they maintained their seven-point advantage going into the interval.
Munster made a flying start to the second-half and immediately piled the pressure on Cardiff Blues.
They kicked a penalty to the corner and launched wave after wave of forward attacks around the fringes. But the travelling defence was resolute and some fine work from Kristian Dacey earned a crucial penalty.
Munster remained camped in the Cardiff Blues half for the next quarter of an hour and Keatley cut the deficit on 54 minutes after his pack won a scrum penalty.
He kicked a further penalty to the corner and again they went through the phases with a succession and pick and drives.
And despite some heroic defence they eventually claimed a try as Dave Kilcoyne burrowed over to give Munster the lead for the first time with Keatley converting.
Anscombe saw a long-range penalty, which would have seen Cardiff Blues draw level, drop short on 66 minutes and that was his last act as Steve Shingler was called into action.
Two minutes later they claimed a sensational try after Matthew Morgan fielded a high-ball and offloaded to release Lee-Lo.
The Samoan centre burst through a hole and released Lloyd Williams with a fine offload of his own. Williams then drew Conway to send Fish clear.
Shingler bisected the posts to give the Blues a four-point lead going into the final ten minutes.
The tension continued to bubble in this thrilling encounter as Keatley responded with another penalty.
But the Cardiff Blues held out in the final minutes of the game to claim a crucial away victory.