Galway scored 1-2 in stoppage time at the end after surviving a nervous finish when they were just two points in front.
Joyce was one of the few Tribesmen to master the heat and the heavy tackles as Joe Kernan’s men breathed a huge sigh of relief as they saw off the dogged challenge of their hosts, whose hopes collapsed when they had two key men dismissed in the second half.
Kernan will be far from happy with Galway’s close shave, although several key players will return by the time they take on Sligo or Mayo at the end of June.
When Nicky Joyce and Gary Sice combined to set up Joyce for a goal after just two minutes, it seemed as if the Tribesmen would romp home.
But by half-time Galway were lucky to be ahead by a point after New York dominated in midfield thanks to the displays from the Kerry pair Pat Madden and Adrian O’Connor.
Galway struggled to cope with the stifling heat and would have been in trouble were it not for the superb accuracy of Joyce.
The former Footballer of the Year, back from injury for his first game since last summer’s championship exit to Donegal, scored 1-3 in the opening half.
But only he and cousin Nicky managed to find the range in the opening half for the Tribesmen and New York hit seven scores to Galway’s six by the interval.
Nicky Joyce pointed a free and added one from play but it was Páraic who rolled back the years with an outstanding display as he built on his goal with three superb points from play.
New York, who did not stand on ceremony in the physical exchanges, never let Galway out of sight, with Joe Kernan’s fellow Crossmaglen native Johnny Murtagh kicking a couple of frees and a sideline.
Skipper Dan Doona from Kerry and Dublin’s Ronan Caffrey also found the range and they were unfortunate not to be ahead at the interval but Benny O’Reilly from Sligo saw a great goal chance flash over the bar to leave it 1-5 to 0-7 at the break.
Páraic Joyce added a pointed free within a minute of the restart but the impressive O’Connor took a pass from Murtagh to respond almost immediately.
The hard-working Gary Sice added a great point for Galway but again the New York response was immediate, with skipper Doona reducing the deficit to just a point again after 42 minutes (1-7 to 0-9).
Páraic Joyce added a pointed free before New York suffered a big blow when midfielder Pat Madden, booked in the 15th minute, was sent off after 48 minutes when he picked up a second yellow card.
And worse was to follow five minutes later for New York when the influential Murtagh picked up a second yellow card for a late tackle on Finian Hanley and was dismissed.
Páraic Joyce pointed the resultant free and Galway fans found further relief 13 minutes from the end when Nicky Joyce pointed a free from the right to make it 1-10 to 0-10.
But New York just would not give up and Dan Doona reduced the margin on the hour mark with another free.
Doona managed two more points for New York but Galway responded to them with efforts from Páraic Joyce and Cormac Bane before Matthew Clancy made it safe in stoppage time with a point to put four between them.
Scorers for Galway: P Joyce (1-8, 0-4 frees); C Bane (1-1); N Joyce (0-3, two frees); G Sice (0-1).
Scorers for New York: D Doona (0-4, one free, one sideline); J Murtagh (0-4, two frees, one 45); J Bell (0-1); R Caffrey (0-1); A O’Connor (0-1); J Bell (0-1).
GALWAY: E O Conghaile; K Fitzgerald, F Hanley, A Burke; D Blake, G O’Donnell, D Burke; B Cullinane, N Coleman; G Sice, F Breathnach, J Bergin; E Concannon, P Joyce, N Joyce.
Subs: M Clancy for Breathnach, (ht); T Fahy for D Burke, (ht); C Bane for Cullinane, (38); D O’Neill for Fitzgerald, (42); D Cummins for Sice, (64).
NEW YORK: P Ryan; C McCarron, B McGourty, R Maguire; A Raftery, A Power, J Bell; P Madden, A O’Connor; J Kelly, P O’Hara, J Murtagh; B O’Reilly, D Doona, R Caffrey.
Subs: K McGeeney for Caffrey, (21); J Moynagh for Bell, (46); R Donovan for O’Reilly, (52); J Killeen for O’Hara, (61).
Referee: M Condron (Waterford).