Former Boro loanee Adam Hammill put the home side ahead after just three minutes but Adomah rounded the keeper to level.
Then after Nahki Wells lofted in a second for Town, Adomah again squared it up with a cracker as Boro started the second half on fire.
Boro made one change to the team that lost 3-1 at home to QPR at the weekend, Emmanuel Ledesma coming in for the injured Kei Kamara.
Huddersfield scored with their first attack after three minutes as a poor touch on the turn by Nathaniel Chalobah in the centre-circle ran free for Oliver Norwood.
He quickly pushed the loose ball to the edge of the box for former Boro loanee Hammill to break free and slot a low shot inside the near post.
Boro responded with a good chance as Adomah barged through two defenders to get to the byline then cut it back to Jacob Butterfield as he arrived inside the box, but his poor shot soared high over the bar.
And Boro levelled on 10 minutes as Adomah pounced on a mistake in the Huddersfield defence.
He darted through to cut out a poor back pass from Gobern on the edge of the box then skipped around the advancing keeper and slotted into an open goal.
Huddersfield had the ball in the net on 15 minutes as Wells streaked on to a ball behind the Boro defence to slot home but the offside flag was already up.
Straight up the other end Boro threatened as left-back George Friend overlapped then squared into a crowded box for Adomah, but his snap-shot was charged down. Then Friend played a one-two with Adomah on the left before his cross was deflected behind for a corner that was nervously scrambled away.
And Boro were on the attack again on 25 minutes as Jozsef Varga won a loose ball on the right and Ledesma squared for Butterfield to drill well over from 25 yards.
Huddersfield won a corner on 28 minutes after a good tackle by Friend on Ward, and when Hammill dipped it to the far post skipper Clarke powered a header over. The game continued to be stretched and open and littered with mistakes with neither side able to impose themselves.
Huddersfield got a second goal with a great strike on 35 minutes.
Adomah conceded a cheap throw and when it was quickly taken down the right it was touched on to Wells just outside the box and he dipped it sublimely over the keeper and inside the far post.
Boro levelled with a screamer 64 seconds after the restart as Adomah again cut out a stray pass from Norwood, took a touch and then absolutely rammed a 20-yard piledriver straight down the middle.
The fired-up visitors piled on a lot of pressure in a high-tempo start to the second period, with a couple of testing corners and crosses forcing the home side into some desperate defending. Boro went close again on 51 minutes as a Ledesma cross was whipped to the far post but Graham couldn’t get a good contact on it and the header flew wide.
Boro threatened on 65 minutes as a poor pass out of the Huddersfield box fell to Tomlin, who took a touch then slammed in a 20-yard shot that came back off the ribcage of skipper Clarke.
Boro had a penalty shout waved away on 78 minutes after Clarke bundled over Graham in the box with a sliding tackle as the striker shaped to shoot but, despite loud appeals from visiting players and fans, the referee played on.
At the death there was a scare as Konstantopoulos dropped a corner into a crowd, but it was hooked clear by Adomah.