Ryan Babel scored the only goal as the Reds beat the Turkish Cup winners 1-0 in the first leg of their Europa League play-off at Anfield.
Victory would have been much more comfortable had Joe Cole converted a second-half penalty and debutant Christian Poulsen not had a goal disallowed.
"I think with our second-half performance we can probably count ourselves unlucky," said Hodgson, who made seven changes and gave captain Steven Gerrard a night off.
"We probably did deserve the extra goal which would have made the trip to Turkey more comfortable. Now it won't be comfortable at all.
"If you miss a penalty you can't blame anybody but yourselves but I thought we were unlucky with the disallowed goal.
"Overall I am very pleased with the result and performance, especially second half, but whether it will be good enough to take to Trabzonspor, only time will tell."
Cole's miss continued a miserable week for the former Chelsea midfielder, who was sent off against Arsenal on Sunday and found guilty of speeding on Wednesday.