Stead pounced from close range after Brett Pitman's effort rebounded off a Leicester upright to settle a topsy-turvy game in the home side's favour and move them five points clear of the relegation zone.
The visitors took the lead four minutes before half-time. Lloyd Dyer seized on a moment of hesitancy in the hosts' defence to lash a fierce drive into the left-hand top corner.
Kalifa Cisse missed a golden chance to level seconds after the restart, dragging his sliding effort wide, but he was on target before the break - shooting beneath Kasper Schmeichel from Stead's throughball.
After the break the home side edged in front, with Pitman collecting Stephen Pearson's pass to beat Schmeichel.
The Robins could not hold their lead, and Neil Danns bundled Tom Kennedy's cross beyond David James to level.
But the hosts rallied once more and Stead secured three valuable points in their battle to avoid the drop.
Pitman, enjoying a rare start up front, almost put the home side ahead after 12 minutes but his powerful drive was punched to safety by Schmeichel.
Despite having the better of the chances in the first quarter, City almost shot themselves in the foot after 28 minutes.
A harmless long ball seemed to be drifting through to James but Stephen McManus nodded a headed backpass over his own goalkeeper and narrowly wide.
The match appeared to be meandering to half-time goalless, but out of nowhere the visitors grabbed the advantage.
Liam Fontaine and Louis Carey stumbled into each other on the edge of the area, the ball broke kindly for Dyer and the winger hammered home from an acute angle.
The hosts should have been level moments later.
Stead headed down neatly for Pitman, whose miscued volley found Cisse unmarked and six yards from goal, but the midfielder slid his effort wide.
However, Cisse made instant amends for his error, equalising seconds before the interval.
Stead slipped a neat pass into the feet of the onrushing Malian and he produced a much more composed finish to slot beyond Schmeichel.
Ten minutes after half-time, the Robins had turned the game on its head.
Pearson latched onto a loose ball 30 yards from goal and fed Pitman, who beat the offside trap to lift the ball over Schmeichel.
James did well to deny Beckford low at his near post as Leicester went in search of a quick reply and the former England stopper was on top form again after 62 minutes to prevent David Nugent from levelling.
The hosts sat deep in their own half for much of the final 20 minutes, inviting and absorbing pressure throughout.
Leicester equalised 14 minutes from time when Kennedy's cross from the left found Danns and the midfielder squeezed the ball beneath James.
No sooner had the Foxes found parity, however, than they were back behind once more.
Pitman found space in the area and, when his deflected effort bounced back off an upright, Stead was there to tap home.