The top scorer netted his 15th goal of the season as the Town made it seven away league matches unbeaten in the second tier for the first time since the 1994-95 campaign, to move up to fourth in the table.
Manager Rob Edwards made two changes to the side that beat Swansea last week. Coming into the team from kick-off for the first time since the draw at Preston last month was Fred Onyedinma, who replaced Alfie Doughty.
Pelly-Ruddock Mpanzu also returned to the side after featuring in last week's victory from the bench. The 28-year-old appeared in midfield for the 360th time in a Luton shirt in place of Jordan Clark, who like Doughty, missed out through injury.
Town started positively and after enjoying plenty of possession, created the chance for Morris to get to the left byline and stand up a cross that was just too high for Allan Campbell as the Scot burst into the six-yard box.
Campbell and Amari'i Bell both had forays into the Blades' box as Town pressed high in the opening exchanges, with Elijah Adebayo tracking back to break up rare United attacks and win possession back.
Ethan Horvath had to field a far post header from Jayden Bogle, who became the first player into referee Leigh Doughty's notepad for a foul on Mpanzu midway through the half as the Town midfielder looked to get a shot off 25 yards from goal after winning a perfectly timed challenge on Berge in the centre-circle.
The late challenge signalled the end of the DR Congo international's afternoon, having to be helped slowly off by two members of the Town medical team, with Luke Berry on in his place for his 154th appearance for the club.
Town were constantly winning the ball back whenever the Blades won possession, and with groans rolling around the stadium, the Hatters worked the ball wide for Gabe Osho to cross, but Berry couldn't keep his volleyed effort down after Adebayo had helped the ball on ahead of Morris.
Anel Ahmedhodzic saw a header saved comfortably by Horvath as half-time approached, but that was the only effort on target in a first half that never truly caught fire.
It certainly had for the Hatters within eight minutes of the restart.
After a patient build-up on the right with Campbell, Drameh and Adebayo keeping the ball before the latter nutmegged John Fleck and raced into the box to cross low for Morris to nip in front of Ahmedhodzic and keeper Wes Foderingham to steer in his third goal in his last four matches.
Morris curled an angled 20-yarder wide before the hosts bared their teeth with 20 minutes to go when Oli McBurnie beat Bell to the byline and crossed deep towards captain Billy Sharp, on as sub, but the Blades' talisman couldn't get the ball out of his feet and Town cleared the danger.
Morris' work was done by the 76th minute, the top scorer leaving the field to a standing ovation from the 1,717 travelling Town fans as he was replaced by Cauley Woodrow.
Berge crossed for McBurnie to head straight at Horvath with seven minutes left, then Edwards brought captain Sonny Bradley on for the last five and injury-time, with Onyedinma feeling the effects of 84 minutes of action.
Sharp fired high over the bar from distance, then McBurnie sent a header straight at Horvath once again, this time from almost on the edge of the area, before Woodrow fired an angled shot just wide of Foderingham's far post with two minutes to go.
The 90 minutes were almost up when Horvath made a crucial point-blank save from Blades captain John Egan in a six-yard box scramble, then McBurnie headed a deep cross wide with time almost up as the Hatters saw out another deserved and brilliant away win with a 15th clean sheet of the season.
TOWN: Horvath; Drameh, Osho, Lockyer (C), Bell, Onyedinma (Bradley 84'); Nakamba, Mpanzu (Berry 26'), Campbell; Morris (Woodrow 76'), Adebayo.
Subs not used: Shea, Lansbury, Muskwe, Taylor.
Goals: Morris 53'
BLADES: Foderingham, Fleck (Sharp 59'), Berge, McBurnie, Egan (C), Lowe, Ahmedhodzic, Norwood (Doyle 69), Robinson (McAtee 70), Bogle, Ndiaye.
Subs: Davies, Baldock, Basham, Clark.
REFEREE: Leigh Doughty
ATT: 28,406 (1,717 Hatters)
