Lewis McGugan's second-half free-kick, a wicked effort from the left, settled the matter and extended Forest's unbeaten home run to a monumental 25 games.
Strachan, who left on Monday after a disappointing start to the season, would have longed for the kind of display that Boro produced in the first half as they did more than enough to be in control, with Leroy Lita guilty of missing the target with two gilt-edged chances.
Forest had their moments in the first period, with Radoslaw Majewski rattling the woodwork, but it was their second-half showing that brought them the points.
Middlesbrough were a shadow of their earlier selves after the break and rarely threatened, meaning it is four defeats out of the last five, while the hosts were able to record their first win in four.
Caretaker boss Steve Agnew will have been pleased with the start the visitors made as they forced home goalkeeper Lee Camp into two decent saves in the opening 10 minutes.
First Camp palmed wide Scott McDonald's shot from distance before getting down low to well save Lita's stinging effort.
Middlesbrough continued to enjoy the better of the play and they should have taken the lead 10 minutes before half-time when, after a neat exchange on the right, Lita find himself in on goal, but he drove his shot across the face.
For all Middlesbrough's much-publicised woes this season, life has not been much better for Forest, but they could have opened the scoring when Dexter Blackstock put his shot wide from six yards with the goal at his mercy.
The game was seeing its best spell of attacking intent from both sides as Lita missed a better chance to put his side ahead when he shot wide from 12 yards after being teed up by Mickael Tavares, while Forest went even closer when Majewski's curling shot on the move crashed against the underside of the bar.
Forest would have been disappointed that half-time had come as they had finally built some momentum, but they continued in the ascendancy immediately after the break and took the lead in the 51st minute.
A driving run down the left by McGugan was ended by a Tavares foul and the Forest midfielder stepped up and produced a wicked delivery that evaded everyone and sailed into the top corner for his sixth goal in the last seven games.
Forest continued to press, with Dele Adebola's glancing header bringing a save out of Jason Steele, although a lack of composure in the final third ensured they could make no further inroads.
Middlesbrough were unable to force Camp into further action at all throughout the second period, only mounting any pressure in the final throes.