Feeling a change was necessary, Stam introduced McShane to proceedings following his return from a hamstring injury, with Joey van den Berg moving into midfield and providing some protection against the Rotherham forward pairing of Danny Ward and Dexter Blackstock.
And the substitution was a beneficial one, as the away side were soon nullifying United’s offensive play and grabbing a stonghold on the game.
Yet for all the possession they would amass before the break, the chances which followed were largely from distance, with Danny Williams and Swift both sending openings into those spectators behind the goal.
A brave block from Darnell Fisher stopped a Roy Beerens pop-shot making its way towards goal, and from the resulting corner, Swift again took aim from a tight angle, with his curling effort only just wide of the far post with Lee Camp in the Rotherham goal struggling to get across.
For the lack of quality and entertainment the first 45 minutes lacked, the second session certainly made up for.
Yet Royals were hugely thankful to Al-Habsi for keeping the score at stalemate.
He would make two stops in quick succession, with the first worth the admission fee alone. Climbing highest to meet another Halford throw, Richard Wood looked to have nodded the ball home – only to see the Omani quickly readjust himself and claw the ball away with his left hand.
Seconds later and he was racing off his line to stop Ward making it 1-0 when the forward was intelligently played through on goal with the Royals defence flat-footed.
Royals were under serious pressure. Completely against the run of play, however, a slick attacking move through the middle cut the hosts wide open and would result in van den Berg, who instigated the phase of play, winning a penalty.
The Dutchman was tripped in the box by Dael Fry, with McCleary assuming responsibility from 12 yards.
The winger went with power and down the middle, but would see the ball cannon back off a leg of Camp and away to safety.
The contest was now an open one but Reading, having weather the earlier storm, were growing in stature and Yann Kermorgant, who had largely been anonymous up front, soon finding himself in the thick of the action.
But from three, albeit difficult, chances, the Frenchman was unable to find the target.
Beerens and McCleary also fired wide and into the hands of Camp before Swift went even closer in the 77th minute.
Superb close control from the midfielder saw him carry the ball across the Rotherham box, dragging in several opponents, before finding a sight of goal.
It looked liked the England Under-21 international had found the top corner once again, but the ball clattered agonisingly against the crossbar.
Royals continued to knock at the door, but remained frustrated by a stubborn United defence closing in on just a second clean sheet of the season.
Such a thought, though, was ruined by an unlikely source with four minutes remaining.
Only clearing a corner as far as Beerens, lurking on the edge of the penalty area, saw the player clip the ball back towards goal.
The combination of Camp and a team-mate dealt with it in a manner, initially, but with the ball spilling loose, the substitute was on hand to take a touch before prodding high into the back of the net.
Despite the jubilant scenes which greeted the goal, the job was not yet done, with Al-Habsi needed once again to keep the home side at bay.
This was when United worked an opening down the right, with the shot-stopper bravely pouncing at the feet of Halford and then Ward in the rebound.
It was somewhat fitting the keeper had the final say in the fixture, as it was undoubtedly his work between the posts which allowed the visitors to go on and snatch all three points.
READING: Al-Habsi, Gunter, Obita, van den Berg, Moore, Williams, Swift (Evans 89), Kelly (McShane 27), Beerens, Kermorgant, McCleary.
Subs not used: S.Moore, Samuel, Blackett, Gravenberch, Watson.
Booked: Gunter, Swift.
Goal: McShane (85)
ROTHERHAM UNITED: Camp, Fisher, Mattock, Wood, Fry, Halford, Taylor, Frecklington (c), Ward, Blackstock (Forde 74), Newell (Brown 64).
Subs not used: Price, Vaulks, Forster-Caskey, Ball, Belaid.
Booked: Blackstock, Halford.
Referee: Scott Duncan
Attendance: 8,800 (inc 644 away)