Well, looking up the etymology for "crawl" (emphasis mine):
crawl (v.) 1200, creulen, from a Scandinavian source, perhaps Old Norse krafla "to claw (one's way)," Danish kravle, from the same root as crab (n.1). If there was an Old English *craflian, it has not been recorded.
Then, the definition of "clawing one's way":
claw your way (somewhere): to move forwards with difficulty, especially by using stiff curved fingers to remove the things that are in your way
This would likely meet the idea that a "dungeon crawl", while initially thought to suggest a snail's pace, might actually mean "fighting your way" through a dungeon.
The Wikipedia article states that the "first" Dungeon Crawl was written by Gary Gygax. In the 112th issue of Dungeon Magazine, a re-print of Gary Gygax's first "Dungeon Crawl" adventure follows the Wizard Mordenkainen, and his apprentice Bigby as they delve through the Dungeons of "Some weird castle", as he put it. Countless hurdles, conflicts and puzzles had to be overcome in order to achieve their goals (of riches).
While this might not be the source for the origin of the phrase, it might help support the use.
I am having a little issue finding any hard reference to the original application of the word to the series, but until then, this is the best I can come up with