A rare batting collapse in their second innings saw Australia set West Indies 287 for victory, but Stuart Clark captured a career-best five wickets for 32 runs to bowl Australia to a 95-run win.
"It's been a pretty good Test from first day to last," Ponting said.
"It was pretty entertaining. Our best overall cricket was probably played on the first day and the last.
"We spoke at length before the start (on Monday) about how we wanted to attack them, what we needed to do, and how we wanted to execute things if we wanted to win the game.
"I think coming into the final day, the game was in the balance. I do not think any side could really say they had their noses in front.
"I guess if you look historically, 212 was the highest winning total at this ground. We knew we had to start well and not give their batsmen any momentum.
"We knew if we kept it pretty tight and put their batsmen under pressure, we had a good chance of getting wickets along the way."
West Indies crashed to 117 for six at lunch on the final day and never looked like recovering, handing Australia a 1-0 lead in the three-Test series which Ponting believes has fortified his team's resolve to not take their opponents lightly.
"I think what this victory shows us is that if we execute well, we are going to be very competitive in a day's play or a Test match," he said.
"This is what we had to get out of our bowling and fielding on the last day. We had to make sure we were switched on and we were prepared to play some really hard tough Test match cricket.
"I really do not think that for the middle three days we did it. We just played a bad shot when we did not need to do it, or we probably over-attacked with the ball in their first innings, but we were unrelenting on the last day and we did not give any opportunities for them to get themselves back into the game.
"It's a great reinforcement for our group going into the next Test."
Ponting also hailed Clark for his bowling and disclosed that his role in the side is well recognised and respected.
"How could you not admire what he's done," Ponting said. "He's averaged 20 in 16 or so Test matches, and that's a really great achievement, especially when he hasn't bowled with the new ball on a lot of those occasions, and we're playing on a lot pitches like like this, where you just have to probably go against what you do in Australia and find different ways of working batsmen over and getting them out.
"He's been the guy a bit like Glenn McGrath was when he was in the side. Whatever the situation required, you know that he'll come in and get the job done for you.
"Whether it be a seaming pitch or someone like the one in this match. He'll give you what is expected and what the team requires."
West Indies coach John Dyson was clearly disappointed that his side let slip such a great opportunity to upset the Australians.
"Obviously, I was hoping, hoping for a much stronger start to the day," he said. "But they bowled very well.
"There were only a couple of sessions throughout the Test that cost us very dearly. That is the thing about Test match cricket - you can't afford to have bad sessions."
Dyson however, acknowledged that his side accomplished a number of good things in the Test match.
"I think we will take away from the game a lot of positive things," said the former Australia batsman.
"That's what we've tried to do since South Africa last year.
"You can't expect a team to turnaround overnight, and consistently win every game. We recognise that we are playing against the strongest unit in the World, and I think we have put on a pretty good show in this match.
"If we can just improve in a couple of areas, we will put on another good show in Antigua. Who knows, they might feel a bit more pressure.
"It might have been a bit of a wake-up to them, they haven't walked in here and just dominated the match and walked away within three days as I think a lot of people expected might happen.
"No-one likes losing. This performance has made them realise that the Australians are human - and not superhuman. They are just human and they can falter.
"If we can apply enough pressure at the right times and continue doing the good things that we can do, they might falter just enough to give us that opening."
Australia have now won 13 of the last 14 Tests they have contested against West Indies. They hold the Frank Worrell Trophy - symbol of Test supremacy against West Indies - and they have not lost a Test series in the Caribbean since 1991.
The second Test between the two sides starts on May 30 at the Vivian Richards Cricket Ground in Antigua, and the third Test starts on June 12 at Kensington Oval in Barbados.
The Aussies also contest a Twenty20 International and five One-day Internationals.