The goalkeeper, 35, denied Simon Francis and Andrew Surman’s long-distance efforts with two diving stops.
Bournemouth edged a first half which was devoid of any clear-cut chances but full of entertainment.
It was the away side that had the opening effort on goal, as in the opening 30 seconds Danny Graham blasted over after Cherries skipper Tommy Elphick gave away the ball in midfield.
In the 20th minute Middlesbrough’s captain Grant Leadbitter was booked after an off-the-ball tussle with Matt Ritchie.
From the resulting Ian Harte free-kick George Friend nearly glanced his header through his own goal but got away with skimming the top of the crossbar.
Mohamed Coulibaly, who was starting for the first time since the opening day of the season, skilfully found a way past Jozsef Varga, but Andrew Surman’s shot was well blocked.
In the 39th minute Lewis Grabban caused the first half’s main talking point as he rounded Konstantopoulos, but his effort was ruled out as the referee claimed he fouled the goalkeeper.
Lee Tomlin had a good chance to give the visitors a lead at the break, but his volley from the edge of the penalty area flew well over Lee Camp’s goal.
The final shot of the first half fell to Andrew Surman, who shimmied his way into space on the 18-yard line but his shot did not test Konstantopoulos.
Ritchie had the second half’s first opportunity as his mis-hit cross struck the outside of the post from the right wing.
The wideman was heavily involved in the 51st minute as his fantastic cross found Lewis Grabban but the striker’s effort was tamely headed towards goal.
On the other flank Coulibaly was also causing problems for the Middlesbrough defence, jumping over three tackles before Surman’s shot deflected narrowly wide, having wrong-footed Konstantopoulos.
Eunan O’Kane thought that he had won the Cherries a 73rd-minute penalty, but he was booked for diving over the defender’s leg.
A minute later substitute Marc Pugh picked up a free ball in the six-yard box but took too long to fire his shot away, while at the other end Graham’s low shot could only trickle through to Camp.
In the 77th minute Simon Francis forced Konstantopoulos into his first top save, as the right-back’s rasping 25-yard effort was destined for the top corner of the goal.
The Greek shot-stopper produced another eye-catching save in the 86th minute after Surman worked a left-footed shot on the edge of the area.
Middlesbrough midfielder Mustapha Carayol left the pitch on a stretcher after stepping on the ball, causing a four minute hold-up as he was treated.