After Ryan Loft's early penalty miss, George Taft headed United into the lead before half-time.
Barrow looked to force an equaliser in the second half and saw a penalty saved by Rory Watson, who made numerous other stops, but levelled in the 80th minute through Robbie Gotts.
Assessing the game, Cox said: "They had a lot of possession, and 16 shots while we had ten.
"in the last 20 minutes they had nothing to lose and could play overloads and leave it one versus one at the back. We had to be better in possession.
"We came here to get a result, and started well though we missed a penalty early on, and then scored through a good set-play.
"One or two of our lads are feeling under the weather with a stomach bug - Lofty has been ill all day while I had to bring Myles (Hippolyte) off as he was also unwell.
"Beest (Alfie Beestin) came on but he's not very well either, while we also have some staff members down ill as well.
"I'm quite proud of the boys for the way we dug in and showed a little bit of character, because we could have easily crumbled, especially when it went 1-1.
"We played well on Saturday but got nothing out of the game due to the refereeing decisions. We came here and knew we were facing a decent side, we dug in and we got a point.
"Rory was brave (for the penalty save), did his homework with the analysis team and saved it well."
Cox concluded by hailing the 86 away fans who made the long journey on Tuesday night, adding: "I'd like to thank the fans who travelled, who will have seen the hard work and effort from the lads. The boys were running on empty towards the end, but we got a point and hopefully the result can be a turning point."
See much more from Neil Cox and Rory Watson on iFollow Iron.
