Jacobs hit five fours and three sixes in his 31-ball knock and got the chase back on track in a well-crafted partnership with Foxcroft, who made 15.
He took on Shoriful in his final overs to wrest back the momentum while Foxcroft played the anchor role.
Shoriful was Bangladesh’s standout bowler with 3-19.
New Zealand’s bowlers earlier bundled Bangladesh out for 102 in 14.2 overs.
Bangladesh never recovered from a disastrous powerplay that saw them lose three wickets in quick succession.
Opener Saif Hassan started strongly with two boundaries before falling for 16 on the first ball of debutant left-arm spinner Jayden Lennox’s international career.
Nathan Smith bowled Tanzid Hasan Tamim for six and then sent Parvez Hossain Emon back for a first-ball duck, leaving Towhid Hridoy to survive the hat-trick ball.
Captain Litton Das injected some urgency, scoring a brisk 26 off 17, but his dismissal after a long rain break left the hosts in trouble.
Hridoy anchored the middle overs before falling to a Josh Clarkson yorker, and Ben Sears wrapped up the tail with two quick wickets.
Clarkson led the bowling with 3-9, while Smith and Lennox took two wickets each.
Leg-spinner Ish Sodhi became New Zealand’s all-time leading wicket-taker in T20 internationals, surpassing pace bowler Tim Southee’s record to reach 165 wickets.
Litton broke the Bangladesh wicketkeeping record for most dismissals, going past Mushfiqur Rahim’s tally of 62 to set a new mark of 64.
Bangladesh won the preceding ODI series 2-1.
