It was tough to take for Gregor Townsend’s side who had led for much of the match but Biggar’s game management won the day after Finn Russell was sin-binned with around 15 minutes remaining.
With the scores tied at 17-17, Wales pressed and pressed and Biggar had the decisive say, clipping over a drop goal to add to his four earlier penalties.
Scotland last won in Cardiff in 2002 when Townsend was in the No 10 jersey but the Principality Stadium continues to be a graveyard for Scottish hopes.
Expectations were high and the bookmakers had the Scots as odds-on favourites. A smashing Darcy Graham try in the first half raised hopes further but Welsh prop Tomas Francis levelled things with a try of his own just after the half hour.
Russell and Biggar exchanged penalties in a match full of ebb and flow but ultimately Scotland were unable to build on last week’s win over England.
It was fast and loose and very, very noisy and Scotland had to weather an early Welsh storm before asserting some first-half authority.
A rampage by Owen Watkin into the Scottish half had the home crowd on their feet but the roars turned to jeers as Stuart Hogg seemed to get away with one when he knocked the ball from Welsh hands.
Wales took the scrum and the first test for the all-new Scotland front row was flunked as WP Nel was penalised and Biggar took advantage, kicking the first three points of the game.
The Wales fly-half repeated the trick three minutes later when Chris Harris failed to move away in the tackle and the hosts led 6-0 after seven minutes. Wales, who had started so poorly against Ireland last week, looked a different proposition.
Scotland quickly regrouped and Ali Price unleashed Duhan van der Merwe down the left wing, igniting a period of sustained pressure from the visitors which led to the first try.
The Scots pushed and pushed and after 17 phases a neat wraparound saw Sione Tuipulotu play in Russell who spun a wonderful pass out to Graham. The winger showed great strength to get the better of Louis Rees-Zammit and get the ball over for his 11th Scotland try on his 24th appearance.
Russell’s conversion attempt from the corner was off target but the stand-off went on to slot over three first-half penalties.
He won the first himself when Wales flanker Jac Morgan blocked the kick-chase and then quickly followed it up with another when the home side were pinged for coming in at the side.
That put Scotland 11-6 ahead at the midway point of the first half but Biggar reduced the margin with his third penalty after an infringement from Grant Gilchrist.
Russell was continuing to probe and threaded through a lovely left-foot kick to the corner. From the lineout, Scotland won another penalty after Wales came in at the side and Russell edged Scotland 14-9 in front.
The lead wasn’t to last until the interval and the last ten minutes of the first half proved trying for the Scots, in more ways than one. Matt Fagerson had to have lengthy treatment on the pitch and was eventually replaced by Magnus Bradbury in the 31st minute.
At the same time, Wales were exerting a ton of pressure and Scotland were starting to creak. The hosts won a penalty in the Scotland 22 and opted to kick to the corner and, from the lineout drive, home prop Francis burrowed his way over for a try.
Biggar couldn’t convert but Wales had restored parity.
Townsend’s side were conceding a dangerous number of penalties but the try at least meant they kept 15 on the field as the first half drew to a close.
As expected, the Scotland coach changed his front row early in the second half, introducing Rory Sutherland, George Turner and Zander Fagerson.
Four minutes later the visitors crept ahead with another Russell penalty after Wales were penalised for not rolling away following a barnstorming run by Tuipulotu.
It was nip and tuck with neither side able to gain the upper hand. Russell tried to give Scotland some breathing space with an ambitious drop goal attempt from almost 50 yards but it fell well short.
The crowd were imploring Wales to make the breakthrough and they almost did on the 56th minute when scrum-half Tomos Williams clipped a delicate kick over the heads of the Scottish defence. Unfortunately for Wales, the bounce of the ball was just too high for Taine Basham but they didn’t go away empty-handed as play went back for an earlier offence by Turner and Biggar was able to tie the scores at 17-17 with his fourth penalty.
The great Welsh stand-off then sparked six minutes of bedlam inside the Principality. It all began with Biggar’s penalty attempt from just inside the Scotland half. He gave it a mighty thump but the ball struck the crossbar and came back into play. Wales seized on it and pounded the Scottish line. Alex Cuthbert thought he was in for a try in the corner but referee Nic Berry ruled that the winger’s foot was in touch.
As he reviewed the action on the big screen, Berry noticed a knock-on from Russell in the build-up as the Scotland stand-off took the ball away from Welsh sub Dillon Lewis with a one-handed intervention. The Aussie official ruled it was deliberate and yellow-carded Russell and awarded Wales a penalty.
The home side kicked for the corner as they chased a second try but eventually had to settle for a Biggar drop goal as Scotland’s defence held firm.
It turned out to be the decisive score, and there was no way back for the Scots whose winless run in Cardiff was extended to what is now likely to be a 22nd year.
Scorers: Wales: Try: Francis. Pens: Biggar 4. Drop goal: Biggar.
Scotland: Try: Graham. Pens: Russell 4
Wales: L Williams; A Cuthbert, O Watkin, N Tompkins (J Davies 67), L Rees-Zammit; D Biggar (C Sheedy 78), T Williams; W Jones (G Thomas 64), R Elias (D Lake 64), T Francis (D Lewis 59), W Rowlands, A Beard, T Basham, J Morgan, R Moriarty (A Wainwright 58).
Subs: S Davies, G Davies.
Scotland: S Hogg; D Graham, C Harris, S Tuipulotu (B Kinghorn 70), D Duhan van der Merwe (C Redpath 78); F Russell, Price (B White 62); P Schoeman (R Sutherland 44; Schoeman 62), S McInally (G Turner 44), WP Nel (Z Fagerson 44), J Gray (R Darge 62), G Gilchrist, S Skinner, H Watson, M Fagerson (M Bradbury 31)..
Referee: Nic Berry (Aus)