Argentinian midfielder Colace raced into the penalty area in the 59th minute to meet Daniel Bogdanovic's cross and head in a fine winner for Barnsley, who fully merited their fourth home league win of the season.
Middlesbrough, who were without the suspended Leroy Lita and unwell fellow striker Dave Kitson, had taken a fortuitous lead midway through the first half through Justin Hoyte, whose attempted cross flew into the top corner.
But Tykes captain Stephen Foster headed in at the far post early in the second half before Colace's well-crafted winner.
The visitors have now won only twice in 10 games under Strachan, who has endured a bumpy ride since replacing Gareth Southgate at the Riverside at the end of October.
Middlesbrough midfielder Julio Arca had forced Barnsley goalkeeper Luke Steele into a smart save before the visitors broke the deadlock thanks to a huge slice of luck.
Boro defender Hoyte combined with Adam Johnson down the left and then over-hit an attempted cross, which inadvertently flew over a stranded Steele into the top corner from 35 yards.
Iceland international Emil Hallfredsson fired a 30-yard rocket wide for the home side and Colace's 20-yard effort was blocked by Boro defender David Wheater.
Barnsley were back on level terms four minutes after the interval. Hallfredsson mis-cued his corner into the box, but Bogdanovic flicked the ball to the far post where Foster rose to power home high into the roof of the net.
Anderson De Silva's thumping drive was then blocked by Hoyte, while Middlesbrough finally stirred themselves in the 53rd minute when Gary O'Neil's lob from the right-hand edge of the penalty area landed just the wrong side of Steele's crossbar.
All the momentum was with Barnsley though and the Yorkshire side snatched the lead in fine style just before the hour mark.
Hallfredsson pulled the ball back from the by-line and Bogdanovic's fine cross from the left was met six yards out by the charging Colace, who headed firmly passed Middlesbrough goalkeeper Danny Coyne into the bottom corner.
The visitors briefly rallied and went close through O'Neil again when the former Portsmouth midfielder's right-footed shot was well saved by Steele.
Boro winger Johnson then fired tamely wide from 20 yards, but Barnsley did not sit back on their lead, with Bobby Hassell lashing a left-footed effort wide from the edge of the area.