The Reds twice took the lead against City, only for the Premier League champions to respond positively on both occasions.
Looking ahead, Allen has urged his teammates to continue improving as the weeks progress.
"[The match] was pretty frantic at times," he told Sky Sports News. "From our point of view it was brilliant to go ahead twice but disappointing to have been pegged back twice.
"After today's draw we are looking to kick on."
Allen was making his Premier League home debut for Liverpool, having signed from Swansea City earlier this summer.