Jacobs, who is on-loan from Derby, struck after just three minutes and Leigh Griffiths sealed the win in stoppage time to move Wolves level on points with leaders Leyton Orient - who they host on Sunday.
It was Wolves' first victory in four games having taken just a single point from their previous three matches, while defeat added to Crewe's relegation worries and they remain in the drop zone after seeing their three-match unbeaten run come to an end despite a spirited display.
Jacobs got Wolves off to the perfect start when he headed home a left-wing cross from Scott Golbourne at the far post.
Crewe boss Steve Davis, a boyhood Wolves fan who was interviewed for the manager's job at Molineux this summer before Kenny Jackett was appointed, must have feared the worst after that early setback.
But his players stuck to their task well and threatened to salvage a point only for strikers Chuks Aneke and Kevin Hitchcock to be denied by smart saves from Carl Ikeme.
Griffiths then finally ended Crewe's challenge with a fierce drive from the edge of the area to grab his 13th goal of the season.