Spurs will now likely have to beat Rennes at home in their final group game to avoid crashing out of Europe's third-tier competition.
And they can no longer catch Rennes at the top of the group meaning they would progress to the last 32, while the group winners are parachuted into the last 16.
Harry Kane started as one of just two players to be retained by Conte from Sunday's 2-1 win over Leeds in the Premier League, which had appeared to launch Conte's reign.
However, the Italian was given an early lesson into the strength of his squad with an embarrassing night in Maribor against a club founded just nine years ago.
Tomi Horvat put the hosts into a shock lead before Ryan Sessegnon's red card left Tottenham a man down for the final hour.
Kane restored parity 18 minutes from time, but even the introduction of five first-team regulars off the bench in the second half could not prevent Amadej Marosa's stunning stoppage time winner.
Europa Conference League - results - collated
JK Helsinki (FIN) 1 (Tanaka 48) Alashkert (ARM) 0
Flora (EST) 1 (Miller 44) Partizan Belgrade (SRB) 0
Maccabi Haifa 0 Union Berlin (GER) 1 (Ryerson 66)
Slavia Prague (CZE) 2 (Olayinka 12, Kuchta 66) Feyenoord (NED) 2 (Dessers 31, 90+3))
Lincoln Imps (GIB) 0 Copenhagen (DEN) 4 (Johannesson 5, Lerager 7, Boving 63, Højlund 73)
Slovan Bratislava (SVK) 0 PAOK Salonika (GRE) 0
NS Mura (SLO) 2 (Horvat 11, Marosa 90'+4) Tottenham (ENG) 1 (Kane 72)
Rennes (FRA) 3 ( Laborde 9, 39, 69) Vitesse Arnhem (NED) 3 (Huisman 43, Buitink 75, Openda 90)
Kairat Almaty (KAZ) 2 (Vagner Love 23, K. Hovhannisyan 56) Basel (SUI) 3 (Cabral 45-pen, Zhegrova 69, Kasami 73)
Qarabag (AZE) 2 (Psaltis 49-og, Andrade 88) Omonia Nicosia (CYP) 2 (Duris 43-pen, Jordi Gómez 90)
Maccabi Tel-Aviv 0 Linz (AUT) 1 (Horvath 89)
Anorthosis (CYP) 1 (Christodoulopoulos 27) Gent (BEL) 0
Bodo/Glimt (NOR) 2 (Brunstad Fet 25, Botheim 85) CSKA Sofia (BUL) 0
AS Roma (ITA) 4 (Carles Pérez 15, Zaniolo 33, Abraham 46, 75) Zarya Luhansk (UKR) 0
Randers (DEN) 2 (Kamara 68, Piesinger 76) CFR Cluj (ROM) 1 (Deac 72)
Jablonec (CZE) 1 (Kratochvíl 7) AZ Alkmaar (NED) 1 (Evjen 44)