Satchel dropped a really interesting article over the weekend, looking at those offenses that have thrived this year despite below average BABIPs (batting average on balls in play). The general rule of thumb is that roughly 30% of all balls put in play result in hits. The type of ball has a significant impact on this as, for example, line drives tend to fall in roughly 70% of the time.
So far this season there are a number of offenses that have what we might consider a low BABIP (e.g. under .280). In fact, of the 21 offenses with BABIPs <=.280 since 2001 9 are currently playing. This may be partly a function of the sample size, but there is some evidence that BABIP has generally been declining of late.
How much of an impact does BABIP have on run scoring? Turns out, a decent amount.
Between 2001 and 2010, the correlation between runs scored and team BABIP was .51 (r2=.26). If you include this year, the correlation declines to .38 (r2=.13). That decline speaks to how many teams with low BABIPs are scoring a decent number of runs so far this year. So, as Satchel suggested, while having a high BABIP certainly helps a team's chances of scoring runs having a lower BABIP doesn't preclude it.
But why do these run-scoring teams have such low BABIPs?
One hypothesis is that teams that hit lots of home runs might generally have lower BABIPs. Since home runs are positive batted balls in terms of runs, but don't count towards BABIP it may be that high HR teams are bound to have low BABIPs. Looking at the data since 2001, this doesn't appear to be the case.
Correlation between HR and BABIP (2001-2010): r=.008, r2=.0001
Correlation between HR and BABIP (2001-2011): r=.16, r2=.0252
Generally speaking, there is no relationship between home runs and BABIP.
That's not to say that the home runs aren't helping those low BABIP teams this year. Intuitively we would expect that if teams have lower batting averages and BABIPs but are still scoring that is likely a function of making the most of their hits--meaning, the long ball.
The Yankees are third in the league in runs scored and not only first in home runs, but first in home runs with men on base (29). The second closest team to them are the Reds with 21. Additionally, the Blue Jays and Rangers are tied for 6th with 15 home runs with men on base. More to the point, if we restrict the analysis to three-run home runs and grand slams, we find that the Yankees rank 1st with 11. The Rangers are tied with the Rays for 5th with 6. The Blue Jays are tied with five other teams for 7th with 4.
This is far from a comprehensive look at the question, but it appears that there is little relationship between HR and BABIP. if a team is going to get away with a low BABIP while still scoring runs it's likely they will do so with home runs and in particular home runs with runners on base.