When the home side pulled an advancing lineout drive down in the 16th minute, Exeter were awarded a penalty try and wing Onisi Ratave was sent to the sin-bin.
But even though they were a man down, Benetton pulled the score level with the first of two tries from Jacob Umaga, after a thunderous run from Tommaso Menoncello.
Manny Feyi-Waboso showed his finishing prowess to restore the lead for the Chiefs, and Slade added a penalty to stretch the advantage.
But Ratave used his power to cross for his 13th try in 13 Challenge Cup games, and Umaga landed a long-range penalty with the last kick of the half to tie the score up at 24-24.
The blizzard of tries continued at the start of the second half, with Harvey Skinner and Andrea Zambonin crossing for the Chiefs and hooker Bautista Bernasconi going over in between for the Treviso side.
Umaga brought Benetton to within four points with his second penalty just after the hour mark.
And when the Halifax-born stand-off struck again with 14 minutes left, the Chiefs trailed.
But Rob Baxter’s side pulled level again with a long-range penalty from Slade and, after calmly keeping control of the ball as the clock kicked down, they eked out a penalty for offside, which Slade stroked home with his trusty left boot from 25 metres out.
Benetton: Gallagher; Ratave, Menoncello, Fekitoa, Mendy; Umaga, Uren; Gallo, Bernasconi, Zilocchi, N Cannone, Ruzza, Zuliani, Favretto, L Cannone
Replacements: Maile, Aminu, Pasquali, Negri, Fa’aso’o, Izekor, A Garbisi, Marin
Exeter Chiefs: Woodburn; Feyi-Waboso, Slade, Rigg, Ridl; Skinner, Varney; Sio, Yeandle, Roots, Jenkins, Zambonin, Hooper, Vintcent, Fisilau
Replacements: Dweba, Burger, Tchumbadze, Tuima, Worley-Brady, Cairns, Haydon-Wood, John
Referee: Luc Ramos (France)