Stephen Peck blazed a penalty over the crossbar as Flixton battled against the elements.
But Adam Stephens and Rafi Leeker struck in the second half to send the tie to penalties.
And the Premier Division leaders were spot-on to progress into the final where they will play Manchester Maccabi or Miam Miam.
Joint-manager Tony Rosenberg said: "We had to dig in and show real character.
"The conditions were atrocious and it was never going to be pretty. Once you miss a penalty you wonder if it will be your day.
"But we battled on, got the goals we deserved and then Dan Garmen did the rest for us."