Brendan Rodgers says Celtic must improve their performances despite booking their place in the Betfred Cup semi-finals.
Celtic needed two late goals from James Forrest and Moussa Dembele to finally see off a resilient Alloa, whose goalkeeper Neil Parry denied several efforts as the Bhoys dominated the game against their lower-league opposition.
Rodgers' side suffered a 7-0 defeat to Barcelona in the Champions League before losing their 100 per cent start to the Scottish Premiership last weekend and the former Liverpool boss admitted ending their winless run was the focus at Celtic Park on Wednesday.
"The main objective tonight was to qualify," said Rodgers. "We knew it was going to be a tough game for us.
"In the group stages Alloa beat Inverness and Ross County so we knew they had great resilience in the team, great mentality and character to defend and they proved that.
"We had a lot of chances in the first half and didn't take them and of course when you don't take them as the game moves on you can become a little but edgy but we showed great patience in the end and scored two."
Celtic goalkeeper Craig Gordon could have been sent off for a foul on Greg Spence late in the first-half and the home side headed into the break having wasted a number of chances, and Rodgers emerged alone in his dugout during half-time after a frustrating 45 minutes.
"I wasn't pleased at half-time," Rodgers added. "We have to be more clinical than that.
"We had 13 shots on target and probably another 13 which weren't on target so to score only the two goals was disappointing. You have to be more ruthless than that.
"The name of the game was being patient but we needed to be much better. Credit to Alloa to come here and defend the way they did and play the way they did was fantastic.
"We deserve the credit for the patience but performance-wise we will obviously play much better."
Celtic face Kilmarnock in the Scottish top-flight on Saturday before welcoming Premier League leaders Manchester City to Celtic Park next Wednesday.