The draw is Everton's third of the new Premier League campaign, and has left them seventh in the table. Huddersfield, meanwhile, have moved out of the relegation zone courtesy of just their second point of the season.
It was Huddersfield that made the breakthrough in the 34th minute, with Billing guiding a flick from Mathias Jorgensen into the back of the Everton net to shock the home supporters.
Everton had chances to take the lead in the period that followed, but Zouma and Calvert-Lewin both missed from headers as the two teams headed down the tunnel at 1-1.
Pickford was again forced to keep out a long-range Mounie strike early in the second period as Huddersfield - in a 5-4-1 formation - looked for their second at Goodison Park.
Rajiv van La Parra also spurned a half-chance for the visitors, before Calvert-Lewin headed another cross over the Huddersfield goal as the match headed for its final 10 minutes at 1-1.
Everton, who had 11 attempts to Huddersfield's nine during the 90 minutes, were the team pressing for a second late on, but Huddersfield earned a point on Merseyside with a battling performance.
As it transpired, for all of Everton's attacking intent, the Toffees only managed one shot on target during the 90 minutes, and it was Huddersfield who certainly had the better chances on the afternoon.
EVERTON (4-2-3-1): Pickford; Digne (Baines 76'), Zouma, Holgate, Coleman; Schneiderlin, Davies; Calvert-Lewin, Sigurdsson (Niasse 76'), Walcott (Lookman 57'); Tosun
HUDDERSFIELD (5-4-1): Lossl; Hadergjonaj (Durm 76'), Jorgensen, Schindler, Kongolo, Lowe; Diakhaby (Kachunga 60'), Mooy, Billing, Van La Parra; Mounie (Depoitre 92')