The game's key moment came in the 24th minute, when Arsenal's Granit Xhaka was sent off for denying Diogo Jota a clear goalscoring opportunity with a high foot that felled the Liverpool forward just outside the Arsenal penalty area.
The incident necessitated a switch in Arsenal tactics to a three-man defence, with Mikel Arteta bringing on Rob Holding for Eddie Nketiah. It was the away side's second early change, after Calum Chambers replaced the injured Cedric Soares in the 11th minute.
But while Liverpool subsequently dominated territory and possession, they could not register a shot on target until the second minute of second-half stoppage time via Curtis Jones.
Minamino had earlier gone close twice from the left side, first with a mishit cross that flew narrowly wide just after half-time and then, midway through the second period, via a cross-shot that also passed the post at Anfield's Kop end.
Arsenal struggled to create scoring chances, but did go close in the 72nd minute; having been found by Kieran Tierney's cross from the left, Bukayo Saka had a shot saved by advancing Liverpool goalkeeper Alisson.
"We revel against the situation," Arsenal manager Mikel Arteta told Sky Sports. "The players showed great fight, determination, brotherhood. You saw the emotion, we never gave up. Credit to the boys.
"I don't know if the red card inspired them but it took the fight. You need a certain attitude to play in these games and the boys did that. We played the game we had to play, which is not our game. I have not seen it back, they checked it so probably a red card."
The meeting at Anfield was originally meant to be the semifinal's second leg, but the venues flipped when last week's meeting was postponed due to a coronavirus outbreak at Liverpool.
And both teams were missing a number of key players on Thursday due to COVID-related reasons -- Arsenal announced before kickoff that Martin Odegaard has tested positive -- as well as injury and Africa Cup of Nations commitments.
"We would've preferred to have scored the goal instead of them getting a red card, Liverpool manager Jurgen Klopp told Sky Sports. "From that moment it looks like we are under pressure [to score] and we had to change the line-up for obvious reasons [before the game].
"In the second half we made a few changes, I thought it looked better and we had clear situations but it was not good enough. But it's a two-legged game and it's half time. I can't remember when it was 0-0 at half time, wherever I've been, and I thought 'we have no chance'. We will give it a proper go."