Higgins was punished for a sluggish start to the match at the city's Motorpoint Arena, losing the opening three frames despite Baird's highest break being only 71 in the first frame.
Higgins - winner of the Welsh Open in 1999, 2010, 2011 and 2014 - stayed alive by winning the fourth and fifth frames, an 81 more like the form that saw him win the China Championship and Champion of Champions tournament in November.
But the Scotsman was finallly forced to accept a surprise early loss when he missed a red to a centre pocket after Baird had ran in 51.
The Englishman pieced together enough points to scramble over the line a year after he had lost 4-0 to four-times world champion Higgins in the last 64.
But six-times World Championship finalist Jimmy White could not complete a shock win in his opening match as he fell to a 4-1 defeat to Neil Robertson.
2010 world champion Robertson opened with a break of 102 only for White to level at 1-1 with a splendid 102 of his own.
That only prompted Robertson to raise his game as knocks of 111, 54 and 57 secured his progress to the last 64.
Read the full article on eurosport.co.uk: Baird defeats former champion Higgins in Welsh Open first round