The Irons were seconds away from suffering a third home defeat in four league games when they trailed Norwich 2-1 deep into stoppage time.
However, a free-kick from Dimitri Payet resulted in a scramble in the six-yard box following John Ruddy’s deflected punch-out before Kouyate lashed the loose ball home as Upton Park erupted.
Slaven Bilic’s men remain in third place, above Arsenal on goal difference and the manager highlighted the need for his players to match the energy of teams like Norwich, who had taken an early lead through Robbie Brady before Diafra Sakho levelled and then looked to have snatched a late winner following a fine strike by substitute Nathan Redmond.
It is an approach Senegal international Kouyate wants everyone to take on board.
“All the players wanted to win the game, but it was a very difficult game. Norwich played well and with very good organisation,” Kouyate said.
With on loan West Ham winger Matt Jarvis not available, Norwich manager Alex Neil rejigged his starting XI at Upton Park, pushing Brady into a more attacking role on the left in front of Sweden international Martin Olsson.
The 23-year-old Republic of Ireland international made the most of his opportunity, latching onto a stray pass from Irons skipper Mark Noble to put Norwich ahead on nine minutes.
While the Canaries were disappointed not to hold on following Redmond’s fine strike with just seven minutes left, Brady is confident they have enough to compete at this level. “It keeps our good run going, but it is frustrating at the same time,” said Brady. “Hopefully we can keep picking up results to get ourselves safe.
Adrián, Jenkinson, Tomkins, Reid, Cresswell, Noble (Zárate 85), Kouyaté, Lanzini (Carroll 72), Payet, Moses (Obiang 57), Sakho
Ruddy, Whittaker, Martin, Bassong, Olsson, Dorrans (Redmond 69), Tettey, Howson, Hoolahan (O’Neil 85), Brady, Jerome (Mbokani Bezua 64)