Cummings scored twice in 10 second-half minutes, almost snatching an added-time winner, to give his side a dream trip to Anfield and hand Reds manager Jurgen Klopp the replay he was dreading – and potentially ruin their plans for their mid-season break next month.
It left Klopp hugely unimpressed at having another match to fit into what has already been a heavy schedule, especially as it was all so unnecessary having been ahead through Curtis Jones' second FA Cup strike in successive matches and a Donald Love own goal.
Such was his desire to settle the tie he called Mohamed Salah and Roberto Firmino off a strong-looking bench in the final 10 minutes but they had little impact as the momentum was all with the hosts, who could easily have pulled off one off the shocks of the competition.
Klopp made 11 changes after Thursday's Premier League win at Wolves but named first-teamers Dejan Lovren and Joel Matip, returning from injury, alongside Fabinho, Takumi Minamino and Divock Origi, making it a stronger team than the one which knocked out Everton in the previous round.
But in spite of the presence of those more experienced players, it was the two stars of the win over the Toffees who continued to seize every opportunity given to them by Klopp.
Pedro Chirivella, who impressed in midfield last time out, played the incisive through-ball down the inside-left channel for Jones to casually stroke past Max O'Leary, becoming the first teenager to score in consecutive appearances for the club since Raheem Sterling in April 2014.
The 18-year-old has a self-confidence bordering on the cocky but that does not mean he is not prepared to put in the hard work and just moments after his goal he was tracking back inside his own penalty area to dispossess Callum Lang.
But his best work is done at the other end and he next popped in the attacking third to flash a left-footed shot wide.
Minamino, still finding his feet after his January move from RB Salzburg, had two chances for a confidence-boosting maiden goal but just missed a cross from Neco Williams and then snatched at Origi's lay-off and ballooned over.
But instead of the visitors pressing home their advantage, the hosts, who lost captain Ollie Norburn to injury midway through the half, were given chances to score as they began to claw back the midfield battle.
Fabinho, in only his third appearance after a month out with an ankle injury, showed his rustiness by giving the ball away too frequently and Shrewsbury should have capitalised.
Shaun Whalley, put clean through, was denied by goalkeeper Adrian's outstretched foot and then also shot wide while Fabinho's unnecessary concession of a corner in first-half added time saw defender Aaron Pierre find himself beyond the Liverpool back line from the breakdown but the ball would not drop quickly enough for him to get off a proper shot.
They were to rue those missed chances as within 28 seconds of the second half starting they were 2-0 down when Love, under no pressure, tried to turned Williams' cross behind but found the wrong side of the post.
Further unnecessary loss of possession, first from Chirivella and then Matip, handed Shrewsbury two more chances as Adrian made a fingertip save to deny Lang in space on the right and Josh Laurent fired wide.
But eventually they were undone in the 65th minute when Yasser Larouci was adjudged to have fouled inside the area as he raced back to challenge Laurent and Cummings sent Adrian the wrong way from the spot.
Liverpool failed to respond – in fact they got more rattled – and, when Lovren misjudged a long ball forward, Cummings capitalised to tuck a shot inside the post.
The sight of Salah and Firmino coming on in the final 10 minutes said everything about the desperation of the visitors' situation and Klopp's desire to avoid a replay, and Cummings almost granted that wish but headed over.