The 28-year-old clipped home from Patrice Evra's cross before Nemanja Vidic deflected the Frenchman's header off van Persie's free kick for the decisive tally.
"Robin's different class," Welbeck told Sky Sports after the match. "He produces moments in games that others can't do and we're glad to have him on our side. As he's shown today, he put the ball in the back of the net.
The England international also hailed the overall team performance, insisting United must look to win every game as it aims to maintain its lead at the top of the table.
"I think during the season, we need to get three points in every game. It's what we set out to do today and thankfully, it's happened," Welbeck said. "We had our gameplan set up, and I think it worked really well to be honest in the first half. Second half, they got into it a bit more but it was a great team performance, I think we dug in really well."
The 22-year-old failed to score but put a fine shift in up front, hassling the Reds' backline and creating chances for teammates.
"I'm going to work hard in the game and once I get my chance playing on the pitch, that's when I can show what I can do and try my best for the team. That's the most I can do and I'm not going to give anything less than that," Welbeck said.
Daniel Sturridge came on as a halftime substitute for Liverpool and added an attacking impetus that saw the Reds push United all the way to the end, and Welbeck held praise for his fellow England international.
"Studge is a quality player with goalscoring instincts as well," Welbeck said. "So when he came on the pitch we had to set up a little bit different, but I think it was a great team performance all-round and they've got many quality players in their side also, it's not just the one."