Bastareaud touched down twice in the first half at Murrayfield and France was only denied another two tries before halftime by some last-ditch Scottish defending.
Flyhalf Francois Trinh-Duc orchestrated the French attacks and an efficient blitz defense restricted Scotland to just three penalties by Chris Paterson.
"Scotland is a team with a big heart and it is hard to play against them," man of the match Imanol Harinordoquy said. "When we want to win, we know we have to give a big performance. "We won the match in the scrum and in the fight."
France joined defending champion Ireland and England on two points in the standings.
France hosts Ireland next weekend, when England goes to Italy and Scotland is at Wales.
France effectively clinched victory in a first 40 minutes in which it shared possession evenly but enjoyed 73% of the territory.
Bastareaud was playing his first test since lying to local police that he had been assaulted during last June's tour of New Zealand, when he actually suffered the injuries while alone in his Wellington hotel room. Bastareaud had to do community service work as punishment and was excluded from France's matches against South Africa, Samoa and New Zealand in November.
His first try on Sunday came when he crashed over from close range following a missed pass by Trinh-Duc and the flyhalf was only prevented from touching down himself by a tackle in the corner from Sean Lamont.
Thom Evans had already denied Vincent Clerc a try when he turned the winger over to stop him grounding the ball, but his positioning contributed to Bastareaud's second score in the 33rd minute. Trinh-Duc spread the play from a throw on France's right all the way to the other side of the field and the Scottish backs were drawn in, leaving Bastareaud with space to charge into and outpace forward Kelly Brown before dummying the last man and touching down.
"We conceded two soft scores, really," Scotland coach Andy Robinson said. "They've one of the best scrummages in the world. "The French were very canny in what they did. It's a very good French side that controlled the game."
Ireland and England each opened the Six Nations on Saturday with an unconvincing victory.
Ireland showed only flashes of its best in a 29-11 win over Italy, while England took advantage of Wales' indiscipline and survived a late scare before winning 30-17 at Twickenham.
Ireland failed to add tries to first-half efforts by Jamie Heaslip and Tomas O'Leary, and even handed Italy its only touchdown of the game with an uncharacteristic error by Ireland fullback Rob Kearney. But Ireland also started slowly last season before winning its first Grand Slam since 1948 and shares the early lead with an England side that marked the 100th anniversary of international rugby at Twickenham with three tries.
It was only with Wales lock Alun-Wyn Jones off the field for a cynical trip on Dylan Hartley that England overcame a scrappy opening with two tries in less than 10 minutes either side of halftime.