Referee José Martínez took just three seconds to show Gaya a red card after the Valencia captain's two-footed, shin-high lunge sent Elanga flying and left the Sweden international needing lengthy treatment before being carried off on a stretcher.
The tackle triggered fury among Newcastle's players and staff, with roughly 25 players from both sides converging on the incident before order was restored, and William Osula among the first to reach his teammate to check on his condition.
Jaissle and his entire backroom staff rushed onto the pitch as Elanga lay injured, taking charge of Newcastle for the first time since replacing Eddie Howe.
Elanga had been one of Newcastle's brightest performers before the injury, having a goal controversially ruled out for a marginal offside decision earlier in the half. He was eventually able to leave the pitch and was seen walking around the dressing room afterwards, having suffered what is believed to be a contusion to the shin rather than anything more serious.
Jaissle later described the challenge as "shocking" but said he was relieved at the outcome.
"I'm really happy that there are no serious injuries. Anthony, we're happy and it seems to be ok, actually," he said.
Jacob Murphy came on immediately to replace Elanga, with Valencia forced to play out the remainder of the first half a man down following Gaya's dismissal.
Newcastle United made four further changes at half-time, with Aladji Bamba, Osula, Harvey Barnes and Leo Shahar all withdrawn, before Yoane Wissa's second-half brace completed a 2-1 comeback win in Jaissle's first match in charge. Newcastle do not return to British soil before facing Everton at Murrayfield in Edinburgh on Wednesday.