Don Meredith had the most games for the Cowboys with 5+ passing touchdowns, with 3 games.

most 5 passing td games by a dallas cowboys player
Interpreted as:
most 5 or more passing td regular-season games by a dallas cowboys player
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Don MeredithDon Meredith315487266.798313.7327.720.811.4143.5
Tony RomoTony Romo210476572.381212.5406.015.411.5544147.6
Craig MortonCraig Morton210263672.259616.6298.027.812.8142.4
Dak PrescottDak Prescott15212777.829510.9295.018.500.011151.8
Troy AikmanTroy Aikman15284957.13627.4362.010.236.11789.0
Danny WhiteDanny White15153345.53049.2304.015.226.122292.7
Eddie LeBaronEddie LeBaron1591560.015810.5158.033.3213.396.0
StatMuse has game-level data for passing touchdowns going back to the 1950 season.