Barnstaple Town’s first game of the new Southern League season saw them beaten 3-0 at AFC Totton on Saturday.
Totton were in command for much of the first half and scored twice in the final five minutes before going into the break.
Jack Masterton broke the deadlock before Byrne headed in from a corner on the stroke of half-time.
Barnstaple started the second half well, but were unable to change the scoreline before the experienced Craig Feeney added the visitors’ third with five minutes remaining.
Barum manager Dean Edwards said: “I thought we matched them for 45 minutes until just before the break. We let in three sloppy goals.
“The difference was they were ruthless in both boxes and we weren’t. We had 10 corners and never got one head o it. There was a ruthlessness about them that we didn’t have.
“I was pleased with the way we played, we just never looked like we were going to score.
“There were a couple of half chances but there were no meaningful crosses into the box until the last five minutes. They were clinical, we weren’t.”
Barnstaple host Wimborne Town in the first qualifying round of the Emirates FA Cup on Tuesday night (September 22).
Edwards warned his Barum side will need to be on top of their game if they want to make ‘a bit of history’ and go into the next round.
He said: “If we give them the opportunities we gave Totton we could be in trouble, but if we keep our concentration and stick at it I think we’ve got a good chance.
“Anything can happen in a cup match. I don’t think Barnstaple have gone through the first qualifying round in some time. If we get through it will be a massive achievement but for me it’s a 50-50 game.
“They are at a high level and it’s down to us to and make a bit of history.”
Barum aren’t in action this Saturday, as they are without a fixture in the Buildbase FA Trophy.