I don't mean to suggest that the First Crusade was an unmitigated success, but despite inconsistent leadership and constant logistical problems the crusaders seemed to manage to succeed at the critical moments needed to reach their overall objectives.
Specifically there were a number of battles and sieges, Dorylaeum, Antioch, Jerusalem, and Ascalon, where the Crusaders won against the odds to some extent. I'm sure each battle had it's unique circumstances, but was there something else going on that added up to advantages for the Crusaders?
Did the Crusaders have some advantage in quality of equipment or soldiers?
Were the tactics they used outside the experience of the Muslim armies they faced?
I've seen it suggested that towards the later part of the First Crusade their army was battle hardened. How much of an advantage, if any, does that cohesion and experience confer?
Or is it even possible to talk about generalities that cut across the numerous string of events that made up the First Crusade?
A combination of factors.
a) Right after leaving byzantine territory, the crusading armies found themselves on the brink of disaster. At Dorylaeum (July, 1097), the vanguard was surrounded by the Turks and only lived another day thanks to the quick judgement of Raymond of Toulouse and Godfrey of Bouillon, who arrived with the main host in the nick of time. The green troops of 1097 had become by 1099 a hardened force: most of the troops had adjusted to a life of self-sufficiency in food, equipment and horses and took at heart the communitarian, militant, religious character of the enterprise. Feudal loyalties gave way to military necessities - Godfrey of Bouillon began the campaign as head of the Lotharingian hosts - he ended it as leader of all northern french troops, a sign that his leadership had succeeded in creating a true esprit de corps.
b) When crusaders arrived from the West, they were profficient in what historians call "the charge and skirmish" type of warfare. They quickly discovered that Easterners did it differently, by employing light cavalry (usually mounted archers) and the rapid tactics of the feint and ambush. By 1098 they were able to successfully counter such tactics; by 1099 they were using them successfully in the march towards Jerusalem.
c) The political situation was a key factor in their success. After the death of Malik-Shah, the Seljuk dominion had fractured beyond repair. Tutush, the emir of Damascus, had taken all of Syria in 1094; in 1095 his realm split between his two sons, that were more interested in fighting each other than making an united front against the invading armies. Worse yet, the Fatimids took Jerusalem in 1098, even by going as far as offering the Crusaders an alliance. A perfect storm.
d) I would add a fourth reason: true religious fervor. While good, cynical opportunists like Baldwin or Bohemond preferred to maintain power over the realms that they had obtained in 1098 (Edessa and Antioch), the rest of the army took down Jerusalem in a haze of religious symbolism and imagery. "Army of God", "Blessed Journey", "Dead Martyrs" were some of the phrases you can find in the letters that participants sent back home. This militant piety, enforced by pope Urban and their past victories, had achieved wonders at the siege of Antioch and would prove invaluable at Jerusalem.
Source: "God's War" (Christopher Tyerman).