Not sure whether this is helpful -- might be too specific to dlnorm, but you can partition [0, Inf[, especially if you have a good idea of where the peak will end up:
integrate.dlnorm <- function(mu=0, sd=1, width=2) {
integral.l <- integrate(f=dlnorm, lower=0, upper=exp(mu - width * sd), meanlog=mu, sdlog=sd)$value
integral.m <- integrate(f=dlnorm, lower=exp(mu - width * sd), upper=exp(mu + width * sd), meanlog=mu, sdlog=sd)$value
integral.u <- integrate(f=dlnorm, lower=exp(mu + width * sd), upper=Inf, meanlog=mu, sdlog=sd)$value
return(integral.l + integral.m + integral.u)
}
integrate.dlnorm() # 1
integrate.dlnorm(-1.05, 10^-3) # .97
integrate.dlnorm(-1.05, 10^-3, 3) # .998