Wales came into the crucial Pool A clash already missing a host of key figures, while Newport Gwent Dragons winger Hallam Amos and Scarlets duo Scott and Liam Williams were all forced off during the game.
However, it was another Scarlet - scrum-half Gareth Davies - whose late try levelled things up before Biggar's long-range penalty completed a dramatic 28-25 victory and a personal 23-point haul.
Scotland kicked off their World Cup with two games in five days, collecting a maximum ten points in the process and going top of Pool B.
First up was Japan at Kingsholm and a Mark Bennett double - complemented by tries from fellow Glasgow Warriors Tommy Seymour and Finn Russell - earned them a 45-10 win.
And they backed that up with a 39-16 triumph over USA at Elland Road, Warrior Duncan Weir and Edinburgh duo WP Nel and Matt Scott all crossing over to score.
Ireland also made it two wins from two as they beat Romania 44-10 in front of a record Rugby World Cup crowd of 89,297 at Wembley.
Munster man Keith Earls scored two tries to move level with GUINNESS PRO12 legend Brian O'Driscoll as Ireland's record try-scorer at the World Cup.
Ulster's Tommy Bowe also doubled up, while his provincial teammate Chris Henry and Leinster full-back Rob Kearney both scored as fly-half Ian Madigan kicked 14 points from the tee.
Scarlets new boy DTH van der Merwe produced an excellent display for Canada - captained by Ospreys forward Tyler Ardron - against Italy in Leeds.
The winger crossed over for a try and made more metres than anyone else (108) but it was just not enough, as Zebre centre Gonzalo Garcia also dotted down to help the Azzurri to a 23-18 win.
Elsewhere, Ulster scrum-half Ruan Pienaar came off the bench as South Africa beat a Samoa side containing Cardiff Blues new boy Rey Lee-Lo 46-6 at Villa Park.
And Glasgow lock Leo Nakarawa played 80 minutes as Fiji once again produced an impressive display in defeat, losing to Australia 28-13 at the Millennium Stadium.
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