Paul Sturrock's Plymouth staged a classic smash-and-grab raid as Jamie Mackie's second-half winner earned three priceless points in their Championship survival bid.
Boro's efforts are concentrated at the other end of the table - but a fourth defeat in five home games saw them drop out of the top six.
New boss Strachan, appointed on Monday, admits he needs to step up his striker search after Adam Johnson's penalty miss meant the toothless Teessiders drew a blank in front of the frustrated Riverside faithful.
But even after crashing Boro's welcoming party for their new manager, Sturrock backed his former Scotland team-mate to guide Boro back into the top flight at the first time of asking.
The Plymouth boss insisted: "Gordon will get them promoted. He's a good manager.
"They're still high up in the table and they've got some good players. I'm positive he will be promoted at the end of the season."
Former Celtic chief Strachan conceded he needs more time on the training ground with his troops.
And he admitted: "It was a wee bit frustrating I suppose. We had a week to try and find the formula and we came up with it in terms of making chances.
"But the problems were with scoring - that's the hardest part.
"As a unit if you're not scoring goals then you make sure they don't go in at the other end.
"We'll only get there by hard work. We can do something about it on the training ground but we need to finish our chances as we got into some nice positions.
"I think we all know what's wrong - you don't need your full UEFA Pro Licence to see that!"
Strachan also challenged his players to prove they are fit enough for battle ahead.
Just two wins in eight league games has left the Scot questioning the physical condition of some of the squad he inherited from axed boss Gareth Southgate.
He said: "We identified early on that while most of the squad's fitness levels are fine, one or two need to do a bit of extra work - and the programme has been designed with that in mind."
Boro dropped to seventh thanks to Mackie's well-taken fourth goal of the season but they were not without chances of their own.
And Sturrock's men had an amazing escape just after the restart. After a great run and cross from Julio Arca, Kari Arnason stopped David Wheater's close-range shot on the line.
But the danger wasn't over. The ricochet sent the ball bouncing back towards goal off Romain Larrieu, before the Plymouth keeper's desperate dive prevented an embarrassing own goal.
When sub Marcus Bent - making his debut after joining on loan from Birmingham - headed wide from Arca's cross, Boro sensed it wasn't their day.
And the excellent Mackie rubbed salt in the wounds with a cool finish after 64 minutes.
He easily out-muscled Wheater to Gary Sawyer's pass, burst through and slotted past exposed Boro keeper Brad Jones.
There was no way back for Boro as Irish midfielder Shane Lowry led by example.
Gritty Argyle defended heroically and survived Johnson's 80th-minute penalty that skimmed a post after Boro defender Sean St Ledger had been sandwiched in the box.
And there was more bad news for Strachan as Emanuel Pogatetz's injury jinx struck again. The defender was carried off early in the second half with a nasty head injury following an accidental clash with Plymouth's David Gray.
And that was in the Austrian international's first start for more than six months following double knee surgery.
Strachan said: "Pogatetz is on the way to hospital and it doesn't look great.
"It's unfortunate for the big fella. We have mental scars at the moment, he's got physical ones."
Sturrock added: "It went according to plan for us and the lads at the back were magnificent, they seemed to sniff out the danger every time Boro attacked."
But It could all have been different for Boro if, with just 12 minutes on the clock, Marvin Emnes had been sharper.
He was presented with a golden chance to break the deadlock only to delay and allow Lowry to block.
Leroy Lita then tested Larrieu with an angled drive and Johnson fired narrowly over. But Plymouth gave as good as they got and Alan Judge fired an early warning with a 25-yard effort just wide from a well-worked free-kick.
Alan Gow came within inches of handing Argyle the lead with a speculative effort on the hour.
Then Jones was forced into a fine save from Mackie after the striker out-witted Wheater - a trick he repeated soon after to deliver victory for Argyle.
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