The hosts went ahead just before the half-hour mark through a fierce drive from their top scorer Ricardo Vaz Te, only for Angel Rangel to equalise within a minute.
A great strike by Danny Graham seconds after the interval put the Swans in front before Nathan Dyer made it 3-1 in the 54th minute, and although Vaz Te pulled one back in the 65th to give the Tykes hope, Graham settled the matter by scoring another with a minute of normal time remaining.
It was a testing assignment for the Barclays Premier League outfit and Pascoe felt their npower Championship opponents deserved plenty of credit.
"It was a tough game," Pascoe said.
"The conditions were very difficult, but both teams tried to get the ball down and pass and Barnsley deserved to take the lead. It was good from ourselves to respond quite quickly and make it 1-1.
"I wouldn't say we were in control for long periods - Barnsley were superb, with the way they passed the ball.
"We got the goals at the right time and in football, if you do that it just eases a bit of pressure. Fair play to Barnsley, because they put us under a lot of pressure.
"We have been coming to Barnsley for years and you know it is going to be tough. At 3-1 I think our players took their foot off the pedal a little bit and that is the worst thing you can do.
"Barnsley got back into it and for the next 10 or 15 minutes, they were the better team."
In the end, Swansea boss Brendan Rodgers was ultimately rewarded for fielding a strong side, making only two adjustments to his first XI from the 2-0 win at Aston Villa on Monday with debutant goalkeeper Gerhard Tremmel replacing Michel Vorm and captain Garry Monk coming into the defence for Steven Caulker.
Many had anticipated a greater number of changes, but Pascoe stressed the Swans were taking the FA Cup seriously.
"The FA Cup is a big competition," Pascoe said.
"We always want to do well in any cups, and especially the FA Cup.
"It is prestigious and it is another game for momentum, away from home, in which we wanted to do well."
Barnsley manager Keith Hill could not fault his team's commitment, but felt errors on their part had been crucial in the outcome.
"We made mistakes which contributed to us losing that game today," Hill said.
"Once we go 1-0 up, with the strategy we used, we shouldn't be conceding that goal.
"Prior to that they hadn't opened us up too many times and so it is very disappointing, but the players will be forgiven.
"The time of the second goal really hurts and hinders our opportunity to get anything out of the game, because we know how good they are with the ball.
"Really, once they did take the lead, there was only ever one conclusion.
"The players are giving me everything, but they are making mistakes that are costing us games."