The Reds travelled to Russia hoping to better their opening Champions League group stage result, when they drew 2-2 with Sevilla.
But they endured yet another night of frustration in front of goal and were left to settle for a share of the spoils.
Fernando’s 23rd-minute minute free-kick put Spartak ahead before Philippe Coutinho equalised just after the half-hour mark.
Thompson, who enjoyed a playing and coaching career with Liverpool, has laid part of the blame for the draw at the door of goalkeeper Kariu
He told Sky Sports: "They've had two attempts, one from the free-kick which they scored, and one from another shot which Karius blocked. But for the goal, the goalkeeper should do much better.
"Fernando puts it over the wall, yes with pace, but it's not even at the side, it's gone sort of four yards inside the post. I think he moves his feet far too late, so you'd have to say it's a goalkeeping error yet again."
Thompson reckons Liverpool let themselves down in the final third too though.
He added: "I would have to say the final ball at times let them down. Too much weight on it, not crossing it early enough, or putting the ball behind players when they had chances.
"This is a massive two points dropped. I think these were the top pick in Pot 1, and you think my goodness, Liverpool have just absolutely steamrolled them from start to finish.
"They will be kicking themselves in that dressing room, with 68 per cent possession away from home."
