The British number five, ranked 115th in the world and coached by former Wimbledon champion Marion Bartoli, overcame a sluggish start after being broken in the opening game.
Dart recovered to draw level at 4-4 in the first set but Tagger immediately broke again and served out the opener.
The first break point of the second set went to Dart, who took it for 5-3 and then levelled the match, setting up a deciding set.
In the final set, momentum swung repeatedly as the pair traded five consecutive breaks of serve. Dart finally steadied by saving two break points to hold for 5-3 and closed out the match to secure the title.