Farsley's captain Lewis Turner was rushed to hospital in an ambulance following an on-field medical emergency.
The 32-year-old had to receive attention 28 minutes into the game, with Gateshead 1-0 up at the time.
Farsley confirmed the postponement, writing on X: "Tonight's FA Trophy fixture at Gateshead has been abandoned following a serious injury to Celts' player Lewis Turner.
"Lewis has now been taken away via ambulance following an on-field medical emergency.
"We will update supporters on Lewis' condition when possible."
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Meanwhile, Gateshead posted: "Following discussions between both clubs, the decision has been taken to abandon tonight's game.
"Our thoughts go out to Lewis Turner & all involved with @FarsleyCelticFC at this time."
Fans present sent their prayers to Turner on social media.
"Thoughts and best wishes Lewis. Praying for a speedy recovery for you," one wrote on X.
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A second said: "Was so scary to watch. Hope and pray he recovers well.
"Hope all of the rest of the team are OK too as that must be terrifying to see happen to a team mate and captain. Take care."
A number of football clubs also sent their best wishes, including Farsley's neighbours Leeds United.
Luke Hannant had put the home side ahead 20 minutes into the game.
But less than 15 minutes later, the players were in the dressing room.
The referee sent both sets of players into their respective dressing rooms as Turner was treated and later taken to hospital.
National League North side Farsley had travelled to Gateshead for the third round of the FA Trophy.
Gateshead are currently fourth in the National League, while Farsley sit 20th in the league below.