Shrewsbury's Daniel Udoh put the visitors ahead in the 27th minute after he slid in Nathanael Ogbeta's cross to score from close range, but Liverpool winger Gordon found the equaliser seven minutes later with a calm finish.
Brazil international Fabinho scored a penalty before half time, sending Shrewsbury keeper Marko Marosi the wrong way, before Firmino added the third in the 78th minute with a brilliant back-heel effort from close range.
Fabinho completed his double in the 93rd minute, blasting the ball into the roof of the net from a tight angle as Liverpool won with manager Jurgen Klopp back on the touchline following his recovery from COVID-19.
"When you get the first goal against you, so many things can happen but I liked the way we responded," Klopp said.
"It was a brilliant goal from Kaide Gordon. He was really calm and composed. His main strength is his finishing.
"All credit to our Academy that they produce these boys but we had no other line-up available to us.
"We had five young players out there. They all did well. We knew we could play better football and we did second half."