Basa netted the winner two minutes into stoppage time after Digne's missile five minutes from time had cancelled out Wahbi Khazri's opener as Lille moved up to fourth on 56 points with five games left, three points adrift of third-placed Olympique Lyon who occupy the qualifying spot for the Champions League third preliminary round.
Bastia are 12th on 39 points, eight points above the relegation zone.
Later on Sunday (1900 GMT), leaders Paris St Germain, who have 67 points, entertain sixth-placed Nice.
In Corsica, the visitors were the first in action when Dimitri Payet's volley was parried away by Mickael Landreau after 13 minutes.
At the end of a swift counter attack, Khazri unleashed a powerful shot that smashed Steeve Elana's left post.
Khazri, however, was luckier on his second attempt two minutes into the second half as his whipped shot from just inside the box went into the top corner.
But full-back Digne sent a rising shot from 25 metres under the bar to give the northerners a deserved equaliser.
Centre back Basa then latched onto a Benoit Pedretti cross to give Lille a lucky win.
In other Ligue 1 matches later on Sunday, Nancy beat Evian Thonon Gaillard 3-1 to move out of the relegation zone, while Sochaux slipped into the bottom three after a 2-2 draw with Bordeaux.
Reims lost their place in 13th after a 1-0 home defeat to Valenciennes, who leapfrog them to 12th.
Another mid-table clash saw Toulouse lose 1-0 at home to FC Lorient, who move seventh.