This was never the most entertaining of matches, but Darren Ambrose's simple 66th minute finish was enough to separate the two teams at Selhurst Park.
Former Newcastle midfielder Ambrose was left in acres of space inside the Boro penalty area to side-foot past Brad Jones after Neil Danns had flicked on Alan Lee's clever delivery from the right.
It was a goal that proved enough to win the match, but Strachan has every right to be furious at the way his side had found themselves in such a position.
He had already witnessed Marcus Bent somehow side-foot a chance well wide of the mark from 12 yards in the first half, before he was left scratching his head after the restart.
Ten minutes after half-time Leroy Lita decided to run and run when Middlesbrough broke with four of their players up against one Palace defender. The chance went begging, much to frustration of Adam Johnson, Justin Hoyte and Rhys Williams.
There was worse to follow. Gary O'Neil was denied by goalkeeper Julian Speroni after being put clean through by Bent.
And then young substitute Jonathan Franks volleyed into Speroni after latching on to another back header from Bent.
Boro might have had the better chances, but a second successive defeat has left Strachan and his recently-inherited squad further adrift of the Championship's top two.
CRYSTAL PALACE: Speroni; Butterfield, Fonte, Davis, Hill; Ambrose, Derry, Ertl, Danns (Carle 81); Sears (N-Diaye 90), Lee. Subs: Clyne, John, Moses, Scannell, Djilali.
MIDDLESBROUGH (4-4-2): Jones; Hoyte (Arca 69), Wheater, St Ledger; Grounds, O'Neil, Osbourne, Williams, Johnson; Lita (Franks 77), Bent. Subs: Coyne (gk), McMahon, Bennett, Emnes, Yeates.