Although Zimbabwe showed some fight early on day five when nightwatchman Donald Tiripano and first-innings centurion Craig Ervine stood firm for the better part of 90 minutes, once they had fallen to some contentious umpiring decisions the hosts folded quickly.
Sodhi's leg-spin claimed three of the last four wickets to go down, while part-timer Martin Guptill finished with figures of 3 for 11 as Zimbabwe were bowled out for 132 in their second innings at the Queen's Sports Club.
Zimbabwe were all out for 362 in reply before New Zealand declared their second innings on 166 for two.
Zimbabwe started the day on 58 for three, and while their official target was 387 for victory, a face-saving draw was their real aim.
That goal began to look within their grasp when Tiripano and Ervine batted out the first 22 overs of the day, seeing off everything that the New Zealand seamers could throw at them.
Replays confirmed that Ervine had not made contact with the ball as it turned past his outside edge.
Australian Paul Reiffel was the umpire on both occasions, and produced another questionable decision after the lunch interval when Graeme Cremer was given out lbw to Guptill despite a clear inside edge.
Sodhi went on to clean up the tail and finish off the match, leaving New Zealand to look forward to a two-Test series against South Africa that begins in Durban on August 19.
"The way the seamers bowled was fantastic, our batsmen spent a lot of time in the middle and the way the spinners performed was also pleasing."
Despite another heavy loss, Zimbabwe could reflect on an improved effort from the first Test, which they lost by an innings inside four days.
The hosts went into that game after a 20-month hiatus from Test cricket.
"When you don't play enough cricket you can come into a Test match and be shellshocked, and that's what happened," said captain Cremer.
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