The Wales outside half contributed 14 points in atrocious conditions at the Arms Park, against a determined Dragons outfit, who were reduced to 14-men mid-way through the first-half.
The hosts trailed 6-3 at the break despite dominating for long periods but Anscombe chipped away and Dan Fish finished a fine try.
Jarryd Sage crossed to set up a tense finale, but Anscombe displayed nerves of steel to kick the winning penalty at the death.
The Blues were under pressure straight from kick-off as they failed to secure the ball. The Dragons seized upon the opportunity and went through a succession of phases as they pressured the try-line but play broke down with a forward pass.
That allowed Mulvihill’s men to work their way out of trouble and they were soon on the attack but it came to an abrupt halt.
Dan Fish had threaded an intelligent grubber kick along the right wing with Blaine Scully and Garyn Smith giving chase. However Zane Kirchner sent Scully crashing backwards as he struck him with a shoulder at full charge.
Following multiple replays, which displayed Kirchner’s late shoulder connecting with the USA captain’s jaw, you could be forgiven for anticipating a red card but referee Frank Murphy opted for only yellow.
With Kirchner in the bin, Cardiff Blues remained in the ascent with the penalty count against climbing but they were unable to make anything stick in driving rain.
Anscombe booted his side into the lead on 18 minutes but he was unable to add a second on 26 minutes with the heavy wind blowing the effort off course.
The 30mph gusts continued to cause chaos for both sides, with the Dragons making things increasingly difficult however they were dealt a major blow shortly before the hour mark.
With the visitors pressing Dillon Lewis was illegally manoeuvred out of a breakdown by Harrison Keddie and just as he did so, Lloyd Fairbrother aimed a swinging elbow at his opposite number’s head. It left Murphy with no choice but to show red.
It was a colossal blow to the visitors but they were soon in the lead with Josh Lewis slotting an impressive snap drop-goal before adding a late penalty.
Anscombe levelled the scores with a well-struck penalty on 48 minutes and five minutes later they claimed the opening try.
Some sublime handling from Nick Williams and Fish unlocked the Dragons defence and allowed the Blues to surge down the left.
Two phases later some more smart handling put Fish away with Olly Robinson supplying the inside pass and Anscombe adding the extras.
Lewis responded to bring the Dragons within four points but when Robinson locked himself over the ball to earn a penalty at the breakdown, Anscombe restored the seven-point advantage.
However, the 14-men Dragons outfit battled back and found a breakthrough, as Jarryd Sage broke past the defence to cross the whitwash, with Jason Tovey's conversion bringing the scores level.
It looked as if the teams would have to share the spoils on the evening, but a powerful scrum in the final play saw Anscombe step up and making no mistakes from the tee to give his side the victory.