The world number one stumbled through the opening set before producing a dominant display in the second to secure a 7-5 6-1 success over the Frenchman.
Murray initially showed no ill-effects of Thursday's gruelling quarter-final victory over Philipp Kohlschreiber as he won the opening three games but world number 15 Pouille hit back, breaking the Scot twice as he won the next four.
The Briton then reasserted his authority by winning four of the next five games to wrap up the opening set in an hour and eight minutes.
Pouille's disappointment at falling short in the first set seemed to break his resistance and Murray, a runner-up at the tournament in 2012, cruised through the second in just 27 minutes.
Spaniard Verdasco awaits the Scot in Saturday's final after he beat Holland's Robin Haase in Friday's first semi-final.