Earnshaw's strike cancelled out Leroy Lita's early opener, and condemned Strachan to a third winless afternoon since replacing Gareth Southgate last month.
Strachan oversaw successive 1-0 defeats in his first two matches as Boro boss, but the Scotsman did not have to wait long to celebrate the first goal of his reign this afternoon.
Just five minutes had gone when Adam Johnson cut the ball in from the left, and while Forest goalkeeper Lee Camp saved well from Julio Arca, Lita was on hand to stab the rebound home from close range.
Lita played alongside his former Reading team-mate Dave Kitson, but the pair struggled to get much change out of a disciplined Forest defence and the sides would have been level at the break had Brad Jones not smothered Guy Moussi's shot from the edge of the area.
Jones produced an even better save in first-half stoppage time, racing from his line to block David McGoldrick's shot after the striker broke through the middle of the Boro defence.
And the Australian was saving his side again seven minutes after the break, tipping Joe Garner's shot around the right-hand post as the Forest midfielder threatened to score from 18 yards.
Jones was finally beaten with 17 minutes left, and the equaliser was no more than the visitors deserved.
David Wheater handled the ball 20 yards out, and substitute Earnshaw curled an excellent free-kick into the top left-hand corner.
Middlesbrough (4-4-2): Jones; Williams, St Ledger, Wheater, Pogatetz; Arca (Digard 66), O'Neil, Osbourne, Johnson; Lita, Kitson (Bent 57).
Subs (not used): Coyne, Taylor, McMahon, Yeates, Franks.
Nottingham Forest (4-4-2): Camp, Gunter, Morgan, Wilson, Cohen; Garner (McCleary 65), McKenna, Moussi, Anderson; Adebola Blackstock 80), McGoldrick (Earnshaw).
Subs (not used): Smith, Chambers, McGugan, Majewski.