Both sides struck inside the first six minutes as Anthony Gordon's controversial equaliser cancelled out Dominic Solanke's opener for Spurs.
Isak then scored for the seventh consecutive Premier League game to claim a sixth straight win in all competitions for Newcastle.
The Magpies remain in fifth but move level on points with fourth-placed Chelsea in the battle for a place in next season's Champions League.
Spurs' ambition of a top-four finish already looks over after a run of one win in eight league games since a stunning 4-0 victory at Premier League champions Manchester City.
Tottenham sit 11th in the table, but a spirited performance from Postecoglou's stretched squad deserved more as Newcastle were left clinging on in the closing stages.
"I'm so proud, so happy with this group of players the way they played and I'm really angry that they didn't get the rewards they deserved today," said Postecoglou.
"I loved everything. The players were given an enormous task and didn't make any excuses. We deserved to win the game today."
Spurs were without first-choice goalkeeper Guglielmo Vicario along with defenders Micky van de Ven, Cristian Romero, Destiny Udogie and Ben Davies through injury plus the suspended Rodrigo Bentancur.
The loss of stand-in 'keeper Fraser Forster to illness meant Postecoglou was forced to throw in Brandon Austin for his Spurs debut.
Slow starts, particularly at home, have been a major part of Spurs' struggles under Postecoglou but they got off to a flyer.
In front of the watching new England manager Thomas Tuchel, Solanke did his case for international recognition no harm with a fine downward header from Pedro Porro's cross on four minutes.
The lead lasted barely over two minutes, albeit the equaliser arrived in controversial circumstances.
Joelinton blocked Lucas Bergvall's pass with his arm to gift Bruno Guimaraes possession and the Brazilian fed in Gordon to fire home.
However, a VAR review deemed Joelinton's handball was not deliberate and the goal stood.
Isak needs no help to find the net on current form but got it from an under-manned Spurs defence seven minutes before half-time.
Radu Dragusin deflected Jacob Murphy's low cross onto the foot of the Swedish striker, who took his tally for the season to 14 goals in all competitions.
Spurs were dealt a further blow as Dragusin, who had been a doubt due to illness, was forced off at the break, leaving Postecoglou without a single natural centre-back to choose from.
Midfielder Archie Gray and full-back Djed Spence deputised manfully as Tottenham did all the pressing in the second half.
Brennan Johnson blasted off the post after Pape Sarr's effort had been parried by Martin Dubravka.
Despite Spurs' lack of bodies at the back, they did have the luxury of introducing Son Heung-min and James Maddison off the bench in the hunt for an equaliser.
Maddison flashed a curling shot inches wide from a short corner before Johnson and Sergio Reguilon both drilled low efforts across the face of goal.
But Newcastle held out to cap a brilliant festive period for Eddie Howe's men that has propelled them into the running for a return to the Champions League.
"Second half we had to defend really well. I'm really pleased to get over the line and win the game," said Howe.
