The Reds midfielder scored on the stroke of half-time after his flick from Divock Origi's cross bounced back off Ben Mee and fell kindly for the Dutchman to side-foot home.
That came after Ashley Barnes had put the visitors ahead, firing in from Matthew Lowton's excellent low cross on just seven minutes to dampen the mood.
After a substandard first 45, Liverpool improved after the break, taking greater control of the game, and reversed the advantage when Emre Can drove a powerful 25-yard strike into the bottom corner past Tom Heaton.
The home side saw out a tense finale to claim a huge three points which strengthens their hopes of a top-four finish.
Wijnaldum says points more important than performances at this stage
Wijnaldum, speaking to the Sky Sports cameras after the final whistle, admitted that it was "difficult in the beginning" for his side and "especially because they [Burnley] scored a goal in the first couple of minutes."
"That made it harder for us," he admitted, adding that Liverpool "just tried to fight" and "tried to do everything to win the game", which he felt they did "better in the second-half" than the first.
"I must say we are also a bit lucky, in previous games that we played like this we lost," Wijnaldum said. "But in those kind of games we didn't have as big a fighting spirit, today was good."
He praised his team-mates' defensive effort and explained how they "tried everything to keep them from our goal" and said he believed they "did that good in the second-half", calling it "a good victory for the team" overall.
The 26-year-old was asked about whether the performance matters as much as the three points, with it the first time Liverpool have ground out such a victory for a while.
He responded: "At this stage I think it is [more important just to win] because we've had a few bad games and if you earn a lot of points and win a lot of games, you can discuss about the football but in this stage of the season, it's better to collect points. It doesn't matter how, you just have to win the games."
The summer signing - who scored his fifth Liverpool goal with his equaliser, all five of which have come at Anfield - also looked forward to their next clash when Klopp's charges face Manchester City in a huge top-four battle at the Etihad Stadium.
He hoped that they can continue their form against the top sides, having yet to lose to any other team in the top six, adding: "We did it before against the bigger teams so we will try to do it again."
Can hails importance of three points
Wijnaldum was joined by fellow goalscorer Can, netting his fourth of the season as he continues his most prolific campaign in front of goal on Merseyside.
The German told Sky Sports that it was "really important" for he and his team-mates to win "because in the past we have lost games against teams from the bottom."
Can continued: "But today we won and we are very happy we got the three points."
The 23-year-old, whose performances have been mixed this season, revealed that the players could feel the tension from the crowd at 1-1 and said: "You know it was a very important goal from Gini at the end of the first-half, we start against the second-half and we played better."
He declared that the Clarets ensured it was "a very tough game" as he hailed that Sean Dyche's men "are strong" and "good in the air", adding: "They played a lot of long balls and were dangerous."
But Can said Liverpool's squad "believed in ourselves" and "played good" and said that they "had a few chances to score" and "eventually did", insisting they are "very happy."
On his goal, the midfielder added: "Of course I'm very happy I scored, I don't score too much, but I think the most important thing was to get the three points."