  Non-Gaussian correlations induced in CMB power spectra by gravitational
lensing must be included in likelihood analyses for future CMB experiments. We
present a simple but accurate likelihood model which includes these
correlations and use it for Markov Chain Monte Carlo parameter estimation from
simulated lensed CMB maps in the context of $\Lambda$CDM and extensions which
include the sum of neutrino masses or the dark energy equation of state $w$. If
lensing-induced covariance is not taken into account for a CMB-S4 type
experiment, the errors for one combination of parameters in each case would be
underestimated by more then a factor of two and lower limits on $w$ could be
misestimated substantially. The frequency of falsely ruling out the true model
or finding tension with other data sets would also substantially increase. Our
analysis also enables a separation of lens and unlensed information from CMB
power spectra, which provides for consistency tests of the model and, if
combined with other such measurements, a nearly lens-sample-variance free test
for systematics and new physics in the unlensed spectrum. This parameterization
also leads to a simple effective likelihood that can be used to assist model
building in case consistency tests of $\Lambda$CDM fail.
