The Phoenix created other good scoring opportunities before halftime — one to Alex Rufer and two to Roy Krishna — but were unable to convert.
As Melbourne City wrested control of the match, there was a suspicion those missed first-half chances could come back to haunt the hosts.
The visitors needed over an hour to notch their first shot an target but that proved the first of many as the previously unemployed Kurto suddenly found himself the busiest man on the pitch.
The 27-year-old made an excellent close-range save to keep out Melbourne's initial effort from Lachie Wells and went on to pull off at least two more saves when a goal looked the likeliest outcome. The second-half shot count was 12-3 in City's favour.
Wellington struggled to retain possession after the break, often turning over the ball in their own half or hacking it clear under pressure.
A Krishna counter-attack looked the likeliest source of a second Phoenix goal, but he got barely a sniff, well contained by the City defence.
Wellington rode their luck and held on to record just their fourth victory over Melbourne City in 17 attempts. The win lifts them to fifth, just one point behind fourth-placed City, and gets Mark Rudan's side back on track after Sydney FC ended the club's record nine-game unbeaten run on Wednesday.
The Phoenix complete their tough run — playing the top four teams in the space of 14 days — with the most difficult assignment, a trip to Perth to take on the first-placed Glory on Saturday night.
Wellington Phoenix 1 (D. Williams 5)
Melbourne City 0