Colin Nish had opened his Dundee account with a fine header after 20 minutes, but Higdon equalised from close range after 74 minutes before heading home the winner 10 minutes later to deny the hosts a first home win of the season.
The result preserves The Steelmen's unbeaten start to the season but was harsh on a Dundee side that had defended bravely for the majority of the match.
Mark Kerr was handed his Dundee debut in place of Italian defender Davide Grassi, who has undergone a knee operation that will sideline him for six weeks.
Dundee's other new signing Brian Easton had to settle for a place on the bench as the Dark Blues looked to build on their 1-0 win at Hearts before the international break.
Motherwell welcomed back Stevie Hammell from injury and Shaun Hutchinson returned from suspension.
Forward Jamie Murphy was also given a place in the starting line-up as manager Stuart McCall made three changes to the side that defeated Inverness 4-1, with Zaine Francis-Angol, Robert McHugh and Jonathan Page making way.
The Steelmen went into the match looking to take full advantage of Celtic's 2-1 defeat at St Johnstone in today's lunchtime kick-off.
The visitors made a positive start and created the first real chance after eight minutes when Higdon nodded the ball into the path of his captain Keith Lasley but Dundee right-back Gary Irvine made a brave block.
Dundee went even closer five minutes later when Darren Randolph, who made his international debut for the Republic of Ireland against Oman in midweek, pulled off a stunning save to deny a goal-bound Lewis Toshney shot.
However, the hosts did break the deadlock after 20 minutes when John Baird's cross found Nish and the striker gave Randolph no chance with an excellent header into the corner of the net.
Dundee had a half-hearted appeal for a penalty waved away when Nish fell under pressure from Adam Cummins in the box, before Jim McAlister's well-struck volley was kept out by Randolph.
Robert Douglas was called into action for the first time four minutes before the break when he comfortably gathered Higdon's low drive before clawing away Tom Hateley's powerful free-kick.
But the former Scotland goalkeeper was beaten on the stroke of half-time when Higdon rifled a shot goalwards, only for the striker's effort to bounce back off the post to the relief of the home support.
Nicky Law worked himself into a promising position after 57 minutes but his cut-back was well cleared by Matt Lockwood.
Nish, who had required a lengthy period of treatment after taking a blow to the face early in the second half, almost doubled the lead for Dundee when he linked up with Baird after 66 minutes but dragged his shot just wide.
And they were to rue that miss after 74 minutes when Henrik Ojamaa, who had replaced the tiring Murphy, crossed for Higdon, who bundled the ball over the line from close range.
Dundee had a shout for a penalty turned down when Neil McGregor was pushed over in the box before Nish headed wide as Barry Smith's men looked to regain the lead.
But it was Higdon who popped up with the winner with six minutes remaining when he powered home a header from a Hateley corner to earn the visitors all three points.