Substitute Mario Balotelli continued his penchant for late interventions with an 85th-minute penalty to give Liverpool a slender advantage to take to Istanbul in the last 32 of the Europa League.
Nine days ago he came off the bench at Anfield to score a crucial winner against Tottenham and did the same against Besiktas, this time at the Kop end after taking the ball away from team-mates Jordan Henderson and Daniel Sturridge to take the spot-kick.
His goal makes the trip to Turkey next week less uncomfortable than it was looking, although the tie is far from over and they were defeated on their last visit there in the Champions League in 2007.
Progress, however small, in Europe is being made as Balotelli's goal avoided setting a new club record of six matches without victory.
That came about mainly because of Liverpool's failure to get through the group stages of the Champions League after a number of laboured performances.
There was an assumption dropping down to the second tier competition, with form having improved and Sturridge now fit, would offer some respite.
But Turkish Super Lig leaders Besiktas never looked likely of succumbing to a repeat of their 8-0 battering on this ground in 2007 -- still the worst defeat in Champions League history -- with a compact and committed defensive performance.
They had already showed they were capable of good performances on their travels with a draw at Spurs earlier this season and they have lost only one of their last 11 away games in all competitions.
A return to the Ataturk Stadium, scene of their famous Champions League final victory in 2005, now awaits but Rodgers will be hoping his side do not need the sort of heroics shown against AC Milan to reach the last 16.
There were plenty of positives in the game with Jordon Ibe, on his European debut for the club just eight appearances into his Liverpool career, a major one.
The 19-year-old, who won the late penalty, was a bright spark on the right and his pace was a constant concern for Besiktas.
An early burst of speed easily took him past two markers to find Sturridge, also making his European bow two years after joining following ineligibility and injury, but his shot from a narrow angle was turned behind by goalkeeper Cenk Gonen.
Moving slightly less quickly was a pitch invader, who entered the fray from somewhere near the Anfield Road End carrying some sort of banner but he did not hang around long enough to convey the message he wanted and was apprehended in the front row of the Centenary Stand.
However, the short break in proceedings was enough to knock Liverpool out of their stride and with Besiktas getting nine outfield players behind the ball breaking down the Turkish side was difficult.
The middle period of the first half saw Besiktas enjoy their best spell with Vasil Kevlak heading narrowly over from Jose Sosa's corner before Simon Mignolet threw himself low to his left to deny Demba Ba after he out-paced Martin Skrtel on the counter-attack.
The incident had shades of last April when Ba, then playing for Chelsea, capitalised on Steven Gerrard's slip to spell the beginning of the end of Liverpool's title hopes.
On that occasion Mignolet could not stop the Senegal international but his save this time around prompted a response from former Reds defender Jamie Carragher
"Why didn't Mignolet do that when Ba was playing for Chelsea! Staying in his goal rather than committing himself. I know SG slipped but....," he wrote on Twitter, which was a touch harsh considering Mignolet was totally exposed by his captain's unfortunate stumble last season.
In the last five minutes of the half Sturridge and Henderson were just off target with free-kicks either side of a thunderous 40-yard drive from Alberto Moreno which Gonen tipped over.
The Besiktas goalkeeper's fumble of Henderson's cross early in the second half saw Adam Lallana blaze over an open goal from close range, although Can's wayward backpass to Ba almost matched that in the error stakes and he was only saved by Skrtel's recovery.
A double change saw Balotelli and Dejan Lovren replace Philippe Coutinho and Joe Allen in the 63rd minute as Liverpool went to two up front and Balotelli's first act was to drill a free-kick straight at Gonen.
With time ticking down it was looking like another frustrating night until five minutes from time Ramon Motta tripped Ibe and Balotelli did the business again, despite Sturridge and Henderson wanting to take the ball off him.