Roman Bezjak gave the visitors the lead on the stroke of half-time at the Stadio Olimpicio, after the Bulgarians had failed to open the scoring from a ninth-minute penalty.
Lazio had the chance to pull level from the spot five minutes into the second period, but Anderson was denied by goalkeeper Vladislav Stoyanov and Ludogorets went on to claim a 1-0 win.
Reja subsequently revealed the Brazilian midfielder should not have taken the penalty in the first place.
"Before each game I write down two penalty-takers, which in this case was Lucas Biglia and Miroslav Klose," he told Sky Sport Italia . "I called Biglia over and asked what happened.
"It seems Anderson earned the penalty and was swept up in the enthusiasm of the moment, so took the ball.
“He didn’t have the cool head needed in this situation. Biglia should’ve stopped him, but these things do happen and penalties can be missed.
"Naturally we did not expect this result and we’ll now have to go to Bulgaria and try to overturn it."
Asked why he made five changes to the side that lost 3-1 to Catania on Sunday, Reja added: "I made changes because Serie A is important.
"Before the weekend we were four points off the European qualification zone, so I had to favour that tournament.
"I have a squad of 24 players and have to let some of them rest."
Lazio's Luis Cavanda and Ludogorets midfielder Svetoslav Dyakov were both dismissed in the second half of Thursday's game and are therefore set to miss the return meeting.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.