The Potters had beaten their Israeli opponents 3-0 a fortnight ago at the Britannia Stadium but Maccabi went into the match unbeaten in 12 European games at Bloomfield Stadium stretching back to 2004.
Stoke had much the better of the first half without managing to break the deadlock, but Whitehead put that right six minutes after the break with a neat finish from a tight angle before substitute Crouch tucked away the midfielder's cross 13 minutes later, with Roberto Colautti's late header just a consolation for the hosts.
Stoke boss Tony Pulis made nine changes from Monday's 3-1 defeat by Newcastle in a week that brings three games in less than seven days, but it was by no means a reserve side.
Kenwyne Jones, Thomas Sorensen, Robert Huth and Matthew Upson were among those who came in along with Danny Higginbotham, making his first appearance after seven months out with cruciate ligament damage.
Maccabi coach Motti Iwanir, meanwhile, gave a start to 18-year-old goalkeeper Barak Levy, and he was given an immediate taste of the challenge facing him when Huth rose alone to head a Whitehead corner towards goal, but placed it over the bar.
Stoke could not maintain their bright start, though, and the next 20 minutes brought only a Jones shot sliced wide after he had beaten the Maccabi offside trap.
The Israelis were certainly giving a better account of themselves and Sorensen was called into action for the first time in the 29th minute when a free-kick allowed Moshe Logasi to fire in a shot, but the 22-yard effort lacked pace.
Stoke had two great chances to go ahead in the latter stages of the first half, the first in the 32nd minute when a shot from Ryan Shotton deflected off Jones.
Levy thought it was going wide and was extremely grateful to see the ball bounce off the inside of the post and into his arms.
Two minutes later a Jon Walters header from a Whitehead free-kick drew an acrobatic save from Levy, with Shotton unable to convert the rebound.
The rain that had soaked the ground prior to kick-off returned for the start of the second half, although not with the same intensity, and the pitch began to cut up badly.
Such conditions are much more familiar to Stoke, of course, and in the 51st minute the visitors took the lead, much to the delight of their small but vocal band of supporters.
Salif Diao lofted a ball into the penalty area and, although Walters was unable to collect it under pressure, it ran nicely for Whitehead to run on to and slide the ball beneath Levy from a tight angle.
It was the midfielder's first goal since January and only his fourth in 97 appearances for the Potters.
Maccabi immediately went in search of an equaliser, with Logasi and Moussa Konate both firing in shots, but all the hosts' opportunities were coming from long range.
And their proud home record was even more threat when Crouch netted Stoke's second after 64 minutes, turning in a Whitehead cross three minutes after replacing Walters.
The striker almost doubled his tally five minutes later with a smart turn and volley from the edge of the area that whistled past the post, while at the other end Konate shot straight at Sorensen.
Maccabi's best chance of the night fell to substitute Colautti in the 74th minute but, 12 yards out, he was unable to dig the ball out from under his feet and stabbed a tame shot at Sorensen.
The home fans were given slight hope when the same player pulled a goal back in injury time with a header, but Stoke saw out the final moments to take their European record this season to seven wins and one draw.
There is still work to be done, though, thanks to Besiktas' win over Dynamo Kiev on Thursday night, although a point against the Ukrainians at the Britannia Stadium on December 1 will be enough.