The £950,000 acquisition from Standard Liege endured a troubled start to his City career after being detained by immigration officials on his arrival and then picking up a hamstring injury.
But all was forgotten four minutes before the interval when the Guinea international collected a pass from Hannes Sigurdsson and drove a low shot past Ben Williams from 18 yards.
The only negative for relieved Stoke boss Johan Boskamp came in stoppage time when his skipper Michael Duberry was sent off for elbowing Crewe substitute Pavel Suhaj.
Stoke had set off at a cracking pace and Bangoura soon endeared himself to the fans by going close in the fifth minute.
He controlled Sigurdsson's cross in an instant and fired a volley which flew a foot over the crossbar.
Bangoure also narrowly failed to convert Paul Gallagher's whipped cross shortly afterwards, but Crewe soon started to show they hadn't come just to defend.
Luke Varney squandered an excellent opportunity after he outmuscled Carl Hoefkens to a long ball from Kenny Lunt in the 15th minute.
The Crewe striker looked odds on to score, but lacked composure and crashed his shot straight at Steve Simonsen.
The visitors also had appeals for what looked a clear penalty refused when Marlon Broomes appeared to punch the ball off the line following Lunt's 34th minute free-kick.
But it was Bangoura who struck to put Stoke on a high and they almost doubled their lead when Darel Russell drove into the legs of Williams a minute later.
Crewe, to their credit, had a real go at the start of the second half and squandered three fine chances to equalise.
Billy Jones headed wide from Lunt's free-kick and Lunt then delivered a corner which Steve Foster nodded over when totally unchallenged.
Crewe came even closer just before the hour when Lunt's shot was half stopped by Simonsen and Duberry scrambled Varney's follow-up effort off the line.
But Duberry was soon at the other end to score the goal which gave his side breathing space and left the Alex with a mountain to climb.
Russell crossed from the right and the Crewe defence was found wanting as Duberry planted a free header inside Williams's right-hand post.
Crewe deserved some reward for their effort, but it just wasn't their night and they were unfortunate to have a late effort by Suhaj ruled out for an infringement.