The central defender met Ruben Lameiras' cross at the far post to secure maximum points and a second-successive win in the Barclays U21 Premier League.
Filip Lesniak and Nathan Oduwa made full debuts at Under-21 level while Luke McGee started for only the second time.
We enjoyed long spells of possession in the first half but Boro, set-up to break quickly, had the better chances.
We started well but only had 20-yarder from Lameiras to show for it. Oduwa and Coulthirst opened Boro up in the left channel but Oduwa looked for another pass and Coulthirst was crowded out.
The home side then sparked into life with three chances in as many minutes.
The first arrived when we lost possession from a quick free-kick, Boro broke and Ryan Brobbel was sent clear, clipped the ball over the advancing McGee only for Gallifuoco to get back to hack off the line.
Birger Meling then got down the left and picked out Lewis Maloney, whose shot looked on target but Gallifuoco was there again to block.
A minute later Maloney squared from the right for Bradley Fewster, who skied the best chance of the half over from 10 yards.
Coulthirst fired wide from distance and Dominic Ball sent Kenny McEvoy away after a run out of defence, McEvoy found Lameiras but he didn't get a shot off.
Mark Kitching made a chance for himself on 32 minutes with a dribble from deep, beating three players on his way into the box before sliding the ball wide of the far post. He then headed wide from Brobbel's free-kick.
Coulthirst ended the half with a snapshot wide after robbing Burn 30 yards out.
The start of the second half mirrored the first as we enjoyed possession with Coulthirst and Ogilvie firing shots straight at Leutwiler. Laste Dombaxe and Souleymane Coulibaly entered the fray for Winks and McEvoy.
Boro hit back again with two quickfire chances. The first saw Brobbel beat McGee to a ball over the top but the goalkeeper made amends by spreading himself to make the save. Jordan Jones then curled inches wide.
Play switched to the other end and Lameiras drew the first serious save from Leutwiler, a low drive from 22 yards that looked to be creeping home until the goalkeeper touched wide. McGee was forced into similar action on 69 minutes, down to his left to deny Brobbel and Leutwiler had to be sharp to turn away Coulthirst's fierce drive three minutes later.
One goal was always going to win it and we struck with eight minutes remaining. Lameiras received a short free-kick in the right channel and floated in a teasing cross to the far post and Gallifuoco did the rest from eight yards.
Middlesbrough: Leutwiler; Halliday, Burn, Weledji, Meling; Jones (Fowler, 82), Maloney (Sirrell, 74), Tinkler (Griffiths, 87), Kitching, Brobbel; Fewster. Unused subs: Okosieme, Fryer.
Spurs: McGee; McEneff (McQueen, 74), Ball, Gallifuoco, Ogilvie; Lameiras, Lesniak, Winks (Dombaxe, 51); McEvoy (Coulibaly, 59), Coulthirst, Oduwa. Unused sub: Priestley.