Mascarenhas belted 22 off the final over as England piled on 65 runs in the last five overs to lift the innings out of mediocrity.
England, trailing New Zealand 1-2 with one match tied, need to win here to square the series.
New Zealand captain Daniel Vettori won the toss and again put England into bat on an easy wicket on the grounds that his side performs best when chasing a total.
It was a tactic which nearly backfired in the fourth match in Napier on Wednesday, when New Zealand needed 341 to win and stumbled in the final overs for the game to end in a rare tie.
But a target of 242 here is not as imposing, and without Mascarenhas and Luke Wright, England looked to be struggling to even reach 200.
Wright top scored for England with 47 off 40 balls, while ahead of him the top-order pace was pedestrian.
Kyle Mills, the pick of the New Zealand bowlers, struck early with the dismissal of Phil Mustard for two in the third over.
Ian Bell and Alastair Cook maintained a run rate of around four an over before Bell was caught by debutant Daniel Flynn for 24 and Cook was trapped in front by Vettori for 42. England were three for 105 after 25 overs.
Neither Kevin Pietersen (39) nor Paul Collingwood (14) was able to lift the scoring rate, and with 15 overs remaining, England were five for 128.
When Mills came back for his second session he took Wright and Owais Shah in the space of four balls.
Wright cracked a full toss straight to Ross Taylor at long-off, while Shah, who had managed a pedestrian 29 off 48 deliveries, was caught by wicketkeeper Brendon McCullum standing up to the fast-medium bowler.
Mills, the pick of the New Zealand bowlers, took four for 36 while Vettori finished with two for 28.