The duo, who came together with the Black Caps still needing 30 runs to avoid the follow-on in the rain-hit Test, put on a defiant stand of 36 after tea on the third day.
Ryder was dismissed for 42, but skipper Vettori returned unbeaten on 33 as New Zealand ended the day at 281-8, still 171 runs behind Sri Lanka's first innings total of 452.
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Tailender Iain O'Brien was the other batsman at the crease on three.
Sri Lanka have two days, weather permitting, to force a win at the Galle International Stadium on a wearing wicket that has already started to assist the spinners.
Just 76 overs were bowled during the day after heavy rain delayed the start by 90 minutes and bad light ended play 20 minutes early.
The Black Caps, led by a gutsy 69 from left-handed opener Tim McIntosh, were sitting pretty at 180-3 when three wickets fell for 15 runs in the post-lunch session.
The collapse was triggered by two wickets from left-arm seamer Thilan Thushara, who had Ross Taylor caug