The Norwegian hosts, overseen by Nilsen following the sacking of Agnar Christensen on Tuesday, went in front in the 65th minute when Zdenek Ondrasek swept home from close range.
Sheriff suffered a further setback eight minutes later when Melli was sent off, but the Moldovan side managed to snatch a point thanks to Ricardinho's volley with three minutes of normal time remaining.
Sheriff started brightly, with Ismael Isa shooting wide on two occasions and Marcus Sahlman saving from Ricardinho.
Tromso's Morten Moldskred could only fire wide inside the box in the 19th minute and moments later, Marcel Metoua did likewise for the visitors.
The exchange of missed opportunities continued up to the interval. For Tromso, Adnan Causevic headed wide a corner by Thomas Drage, who later drilled off-target, and Moldskred shot over, while for Sheriff, Kobi Moyal's long-range effort was tipped behind by Sahlman and Cadu twice put the ball the wrong side of the post.
Henrique Luvannor brought a save out of Sahlman early in the second half and then, with their first meaningful attack since the interval, Tromso took the lead as Ondrasek finished six yards out from Magnus Andersen's pass.
Melli received his marching orders for a foul on Moldskred and in the 85th minute Ondrasek saw his shot caught by Vjekoslav Tomic.
Ricardinho then made his late intervention, taking advantage when Ruben Kristiansen failed to deal with a cross.
In their last Europa League outing, Tromso lost their Group K opener 3-0 at Tottenham, while Sheriff drew 0-0 with Anzhi Makhachkala, who were beaten 2-0 by Spurs on Thursday.