"Iraq beat Australia during its title run in 2007"

This means that Iraq and Australia were the top teams who competed against each other to win the 'Asian Cup'. The 'title run' was the competition, or playoff game to win the title. The Asian title winner then plays other zone/title winning teams for the top prize which is to win the World Cup title.

The 'title' is the prize for the winning team, the 'run' is the tournament between teams to win.

Here is how it is set up: (quoted from the first reference URL)

"Qualifying tournaments in each of the six FIFA continental zones (Africa, Asia, North and Central America and Caribbean, South America, Oceania, and Europe) determine which teams will play in the World Cup games. Competing teams are divided into 8 groups. In 2010 the groups were divided thus:

Group A: South Africa, Mexico, Uruguay and France
Group B: Argentina, Nigeria, Korea Republic and Greece
Group C: England, United States, Algeria and Slovenia
Group D: Germany, Australia, Serbia and Ghana
Group E: Netherlands, Denmark, Japan and Cameroon
Group F: Italy, Paraguay, New Zealand and Slovakia
Group G: Brazil, Korea DPR, Côte d'Ivoire and Portugal
Group H: Spain, Switzerland, Honduras and Chile

These groups play a round-robin tournament, with the top two teams from each group advancing to a second stage, known as the knockout stage. At this point, teams play against each other in one-off competitions, with teams being eliminated until the quarterfinals, semifinals and finals. The losing semifinalists also play to determine third place."

This is how I understand 'title run' which seems to fit given Iraq's successful results that year

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Note added at 2 hrs (2011-01-20 23:48:48 GMT)
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addendum:

"succession of matches that they won, culminating in victory at the final"