The Dons had earlier taken the lead through man of the match Andy Dow who rifled the ball low past Woods from eight yards following fine build up play by Rachid Belabed and Arild Stavrum.
Motherwell equalised in controversial fashion four minutes from half time when Don Goodman went down in the box following a challenge from Jamie McAllister who looked to have played the ball.
Ged Brannan converted the spot kick and, after an even second half which saw chances created at both ends, Solberg netted the winner.
Thanks to results in the First Division, there will be no play-off and the Dons are safe in the Premier Division.