A late Mario Balotelli penalty gave Brendan Rodgers' side a slim advantage from the home leg, and they travel to Turkey in fine form after an excellent run of results.
Sunday's 2-0 Premier League win at Southampton was their sixth triumph in seven games in all competitions, and the men from Merseyside have lost just once in 18 since mid-December.
Although their challenge for a top-four finish and UEFA Champions League qualification appears reignited, victory in this tournament would also see Liverpool return to Europe's premier club competition, regardless of their final league placing.
After losing Luis Suarez to Barcelona in the off-season, Liverpool's campaign got off to a stuttering start, but midfielder Joe Allen believes they are now back on track.
"Sometimes [criticism] gets taken out of hand but I don't think it was unfair," he told the Liverpool Echo. "There was no hiding places and we were not delivering as players.
"There was no excuse from us. We knew there might be a difficult patch early in the season and that probably went on longer than we hoped and expected.
"At the same time we always knew we would come through it and I think that was the biggest sign we would be where we are at this stage."
After the defeat at Anfield, Besiktas fell to a surprise 1-0 defeat at Eskisehirspor at the weekend, meaning they lost their place at the top of the Turkish Super League to Galatasaray.
And manager Slaven Bilic says a long list of injuries and suspensions are hampering his squad ahead of the return leg.
"We've had significant losses in two critical matches," he said. "There are significant [players] missing ahead of the Liverpool second leg but we will do our best to overcome it.
"Losses can be compensated in the league, but this is impossible in Europe."
Goalkeeper Tolga Zengin (knee), defender Ismail Koybasi (knee) and forward Cenk Tosun (foot) are all injured, while Ersan Gulum and Ramon Motta are suspended after picking up cautions at Anfield.
For Liverpool, captain Steven Gerrard remains a doubt with a hamstring problem, which could lead to another on-field debate about penalty duties should the visitors be awarded another spot-kick after Balotelli took the ball from stand-in skipper Jordan Henderson last week.
Mamadou Sakho's hip problem is also a cause for concern after he missed the trip to Southampton, while Lucas Leiva (groin) is doubtful and Lazar Markovic remains suspended.
