Alex Lawless was the matchwinner with a well-taken volley within seconds of the restart after half-time, ensuring the non-league side will be in the hat for Sunday's fourth-round draw.
It was a deserved victory for the Hatters in front of a sell-out Kenilworth Road crowd, with 59 places separating the two sides in the football ladder.
But it only deepens lacklustre Wolves' woes after an awful festive period containing three successive defeats, with Solbakken, whose future has been the subject of much discussion, admitting in the build-up to the cup clash that Wanderers were in crisis.
That crisis has now deepened for Solbakken, appointed in the summer, despite the Norwegian receiving the backing of his players this week, with the Black Country club now boasting just three wins in their last 17 matches.
Nervous Wolves fans went into the clash with memories of their embarrassing 1986 FA Cup exit to non-league Chorley at the forefront of their minds.
And after a positive start from Luton, who were pressing at every opportunity, it was clear the out-of-sorts visitors would be in for a tough afternoon.
Both sides enjoyed clear openings inside the first six minutes, JJ O'Donnell firing over from the edge of the penalty area before Kevin Doyle got in behind but could only send his effort just wide.
Bakary Sako flicked an effort off target for Wanderers before Luton were handed a golden opportunity in the 22nd minute.
Wolverhampton-born Andre Gray, released by the Black Country club as a teenager, found himself clean through but, bearing down on Carl Ikeme, saw his effort well saved by the Wolves goalkeeper.
The lively Gray was fired up for the tie and created another opportunity moments later but could only blaze over from 12 yards.
Anthony Forde was presented with a solid chance just after the half-hour mark but produced an air shot, with Wolves going on to enjoy a rare spell of pressure.
Hatters keeper Mark Tyler was called upon to superbly tip Sako's inswinging corner around a post, before again producing another top save to deny the influential French-born winger two minutes later.
Jon Shaw flashed a good chance across goal and out for a throw-in at the start of the second half, but it was from that throw which Luton notched what turned out to be the winner.
Wolves made a mess of it and a cross was duly sent into the box, Shaw went up for header and the ball fell to Lawless who volleyed into the bottom corner from the edge of the area.
The hosts were firmly on top, O'Donnell drilling over, but Wolves rode the storm and Christophe Berra nodded against the top of the bar at the other end.
Doyle headed straight at Tyler in the 63rd minute but Wolves, devoid of confidence and ideas, were failing to do enough.
O'Donnell was stretchered off after receiving lengthy treatment following an accidental kick to the head by substitute Slawomir Peszko.
And despite nine minutes of injury time, during which Tyler tipped over from Jamie O'Hara, the Hatters ran out winners.
- Sports & Recreation