Britt Assombalonga netted his sixth of the season to earn a point after Monk’s side fell behind at Oakwell.
Ashley Fletcher, returning to face the club which he arrived at on loan in 2016, headed into his own net under pressure from Tom Bradshaw to give Barnsley the lead in the third minute.
Boro responded immediately through Martin Braithwaite (seven) who netted from 10 yards out after Barnsley failed to clear a corner.
Barnsley had their second and the game’s third after just nine minutes when Cameron McGeehan rose highest to head Brad Potts’ deflected cross past Darren Randolph.
The visitors pushed for an equaliser but Barnsley defended stoutly until Stewart Downing unlocked their defence allowing Assombalonga (60) to go clean through to provide the finish.
Monk said: “We’re frustrated in that we’ve come looking for three points and only managed to get one, but it was a frantic opening and unfortunately for us we conceded that first goal.
“It’s hard to criticise the own goal for the first goal, but the second is avoidable and come from our mistake. But we responded well and we totally dominated the game after that.
“We created good opportunities to win the game but we weren’t clinical enough and not ruthless enough.
“So it’s both boxes in the sense of not causing our own mistakes and giving ourselves an uphill battle and then also being ruthless when you’re dominating games like that.
“It’s a difficult place to come and to dominate like that after the first 10 minutes and get into the situations we did are the positives which we’ll take forward into the next game.”
Barnsley head coach Paul Heckingbottom said: “We’re obviously disappointed that we’ve been in front twice and not come away with the win.
“We weren’t at our best with the ball, but did some good bits without it which were pleasing, so it’s a fair result.
“We weren’t good enough with the ball to keep it from Boro for long spells and they weren’t good enough to keep it from us.
“The game had spells where it was really good and had spells where it really frustrated me. But we’ve managed to take a point from Boro so I have to be realistic and think that the players have come a long way.
“They’ve got quality all over their squad so we know we have to be spot on all the time. We had to prepare and because we have a brand-new team this year it’s all new to them and bit by bit we’re getting better.”