David Accam netted for the home side but Abu Danladi's double was enough to ensure a 2-1 win for the visitors.
It took 36 minutes to break the deadlock at Toyota Park, but it was the easiest of jobs for Danladi who tapped in from six yards after being located by Ethan Finlay's cutback.
On the stroke of half-time, the 21-year-old scored again to take his season tally to five.
This time around, Jerome Thiesson sent in a cross from the right and the Takoradi-born was at hand to prod home from six yards.
Accam, however, reduced the deficit on 77 minutes, firing a shot beyond goalkeeper Bobby Shuttleworth after being put through on goal by Bastian Schweinsteiger.
The former Helsingborgs winger has now registered 14 goals in 24 matches this season.
The centre-back headed home to put Crew 2-0 up, connecting to Federico Higuain's free-kick.
Ola Kamara had put the home side in the lead but Walker Zimmerman halved the deficit for Dallas 12 minutes to full-time.
Mensah's compatriot and teammate Mohammed Abu lasted for the entire duration but Harrison Afful and Lalas Abubakar were both second half substitutes.