A crescendo of boos from the home supporters greeted the final whistle, while jubilant United fans celebrated their team rising to ninth in the table in manager Simon Grayson's 100th match in charge.
Boro were widely tipped to launch a promotion push under Strachan's stewardship but find themselves languishing in 20th - one point above the drop zone - after another damaging loss.
Davide Somma needed just 12 minutes to open the scoring and despite Kris Boyd's second-half leveller, Luciano Becchio lashed in a fine volley as United earned a deserved win.
Gary O'Neil shook off a hamstring problem to return for the home side, while Leeds were without the suspended Alex Bruce and keeper Kasper Schmeichel.
The two teams paid tribute to Malcolm Allison with a minute's applause prior to kick-off following the late Boro boss' death this week.
Even though this was the hosts' biggest crowd of the season, red seats were aplenty at the Riverside following the Teesside club's underwhelming start to campaign.
And they began on the backfoot as Jonny Howson powered into the box and drove a shot over the bar, before Robert Snodgrass sent a daisy-cutter into the hands of Jason Steele.
Nicky Bailey miscued a volley horribly wide and the comedy continued when United midfielder Amdy Faye inexplicably caught the ball in the box after mistakenly hearing a whistle from referee Chris Foy.
It took the Whites just 12 minutes to earn reward for the bright opening when Somma swung a left leg at Sanchez Watt's deflected centre to send the ball past Steele's despairing hand.
Scott McDonald and O'Neil fired in shots that failed to trouble Jason Brown as Strachan's players struggled to create, although Boyd should have done better when he evaded the attentions of Neill Collins to head wide.
Somma ought to have played in Luciano Becchio when two-on-one just before half-time, but the striker opted to shoot despite his colleague being unmarked on the edge of the box.
Middlesbrough improved after the break and Boyd found his range to grab an equaliser, getting in-front of Richard Naylor to prod in the 53rd minute leveller from Matthew Bates' inviting cross.
Leeds rallied and they restored their advantage 10 minutes later in emphatic fashion when Becchio sent a stunning left-foot volley in off the post from Bradley Johnson's clever cut-back.
The Argentinian was brimming with confidence and he cracked a shot off the crossbar with Grayson's team threatening to put the game out of sight.
Boro put in a late rally in search of an equaliser but seldom tested Brown as Leeds left with all three points.