Non-league Barnet's adventure in the competition came to an end as they lost 3-1 at fellow Londoners Brentford.
Newport had knocked out 2016 Premier League champions Leicester City in the third round and were set on their way to another giant-killing when Robbie Wilmott struck two minutes into the second half on a soggy Welsh pitch.
Irish striker Padraig Amond doubled the lead 20 minutes later with a magnificent goal as he swivelled to meet a corner and curled the ball high into the net with one touch.
The goal sparked wild celebrations as Newport, 14th in League 2, reached the fifth round for the first time since 1949.
It was an unforgettable night for their goalkeeper Joe Day, whose wife went into labour during the game.
Wolves took an early lead against third-tier Shrewsbury through Matt Doherty but then fell 2-1 behind before Doherty levelled just before the interval.
Portuguese Ivan Cavaleiro struck in the second half to win the tie and set up a fifth-round game at Bristol City.
QPR will next host fellow London side Watford, while Brentford visit Swansea.
(Reporting by Richard Martin)
