The Portugal international lasted just 23 minutes before an awkward tumble saw Andy Carroll introduced as a replacement. Carroll scored one of the Reds' three goals, but Meireles faces an uncertain spell in the treatment room.
"It's his collarbone but we don't know how long he'll be out," Dalglish told Sky Sports. "I think we are going to be without him for a couple of weeks.
"I've not got a medical term for it. He just damaged his collarbone. He just fell on it. It's disappointing. It doesn't matter if he's been playing good, bad or indifferent, it's still disappointing to pick up such an innocuous thing. We are disappointed. But I'm sure Raul is the most disappointed person."
Luis Suarez was the star man once again for Liverpool, scoring his third goal of the season before laying on Maxi Rodriguez's first. Dalglish was typically subtle in his praise of the forward, and he also offered appreciative words to Carroll - who had not scored before Wednesday evening.
"I'm pleased for him," Dalglish said of Carroll. "And I'm pleased for Maxi who also got on the scoresheet for the time this season. Obviously Luis is hogging the limelight a wee bit by scoring again but it's great, especially for Andy to get off the mark with fantastic strike.
"We are delighted for him. He might have come on a bit later had Raul not got injured. But he did well when he came on."
Asked specifically about Suarez's contribution, Dalglish responded: "Well he doesn't give many bad performances does he."
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