Recalled Taranaki batsman Will Young(62) looked to keep the Stags on course, but lost Hay and then Smith cheaply, dismissed for three by a caught-behind off the bowling of Michael Rippon (2-56) in the 19th over.
Hawke's Bay's Kieran Noema-Barnett (76) joined Young for a 51-run partnership to keep hope alive at 182-3 in the 26th over, but Otago then ripped through the middle order, taking four wickets for just 12 runs, courtesy of Dean Foxcroft (2-39) and Michael Rae (2-49).
Brad Schmulian (35) joined his Hawke's Bay team mate Noema-Barnett to get their side through to 269-7 in the 40th over, but with a run rate approaching 14.
Kitchen returned to break the partnership by getting Noema-Barnett out as the last hope, and then cleaned up the tail in two overs.
Smith and the Stags will be hoping for a better performance on Wednesday in the return Ford Trophy game at Nelson's Saxton Oval.