Braintree took the lead in the 33rd minute through Kenny Davis, who converted from the spot after a foul by Joe Skarz, but Oxford were back level just before the interval as Liam Sercombe netted from 12 yards.
Oxford established a two-goal margin with a double from Pat Hoban, and after Davis missed a penalty six minutes from time, the League Two outfit were able to secure their passage into the next round.
Newport County needed just 36 seconds to take the lead against National League North side Brackley Town with Lenell John-Lewis getting the goal, before Yan Klukowski doubled their advantage after 17 minutes.
Six minutes before the break, however, Aidan Hawtin pulled one back for the visitors, but Alex Rodman and an own goal from Hawtin ensured that the Welsh club avoided a scare against non-league opposition with a 4-1 win.
After scoring twice in the first seven minutes through Tom Barkhuizen and Aaron Wildig, Morecambe looked in control of their replay at home to Dagenham & Redbridge, but in the space of 41 minutes, the visitors had completed a stunning turnaround.
Results in full (half-time scores in brackets): Morecambe 2-4 Dagenham & Redbridge (2-2), Newport County 4-1 Brackley Town (2-1), Oldham Athletic 2-0 Mansfield Town (0-0), Oxford United 3-1 Braintree Town (1-1)