Gylfi Sigurdsson's spectacular 13th-minute free-kick was enough to secure Swansea a 1-0 victory but Monk's men had to withstand late pressure at a rain-lashed Liberty Stadium with goalkeeper Lukasz Fabianski making smart saves to deny Christian Benteke and Gabriel Agbonlahor.
"I thought we were the better team and created enough chances to score that second goal and be more comfortable," Monk said.
"As the game drew towards the end Villa had to throw everything at us, but I thought we defended very well before the keeper comes in at the end.
"He (Fabianski) has done well but that's his job
He didn't have too much to do but in the critical moment he has shown his concentration and professionalism."
Swansea opened their festive fixtures by grinding out a 1-0 win at Hull and it was much the same for long periods in the second half against Villa as Paul Lambert's side probed for an equaliser.
"We've had it all season apart from a couple of games where we could have done better," Monk said.
"Our mentality, the way of defending, we've made the players work very hard and what it means to defend as a team.
"Every single player has a responsibility, they clearly understand that and it's nice to see they know what's needed.
"Okay, we would love to have attacked for 90 minutes and play wonderful football but the conditions didn't help both teams.
"We create good chances, goalscoring chances, and we could have been more clinical but I don't feel fortunate at all to have won that game."
Villa's defeat was only their second in eight games and Lambert said he was delighted with his players' performance.
"I'm proud of them for the way we played," Lambert said.
"We probably did to Swansea what they do to most teams coming here
I thought we out-passed them and penned them in at the end and I'm really pleased with the way we're playing at the minute.
"Sometimes you travel home and say you didn't deserve anything after losing, but I'm delighted with way we're playing at the moment."
Lambert said he would not admonish goalkeeper Brad Guzan for allowing Sigurdsson's winner, even if the American felt he could have done better to stop the Icelandic international's free-kick.
"I think Brad will be disappointed with their goal," Lambert said.
"But there's no attachment of blame from me because Brad has made fantastic saves for us in the past and will make them in the future."
Villa had what looked a genuine penalty claim 18 minutes from time when Agbonlahor appeared to be shoved in the back by Swansea skipper Ashley Williams when running through on the home goal.
But Lambert said: "I would have to see it again
If that's the case (a shove) then maybe you think it's a penalty kick, but I would have to see it again.
"I never felt we were in trouble other than the free-kick, football-wise we were excellent in the first and second half.
"I just made a change at half-time bringing Andy (Weimann) on because I wanted another offensive player on and a bit more in the last third."
Source : PA
Source: PA