Runaway Premier League leaders Liverpool looked to be heading into the fifth round for the first time since 2015 after a fine first-half finish by 18-year-old Curtis Jones was followed by Donald Love's own goal in the opening minute of the second half.
Yet Shrewsbury were excellent throughout and were rewarded with two goals in the space of 10 minutes by substitute Cummings.
He beamed: "I could have got a hat-trick! It was a dream come true against the best team in the world. Going to Anfield is what it's all about.
"Anfield will be really good, for the fans, for the chairman making a few quid. We can't wait to get there."