The England international was dropped from the starting line-up in the wake of the 2-2 draw with Leicester City last week but came off the bench to snatch a point for his side at Old Trafford on Tuesday.
United were 2-0 down at half-time thanks to goals from Ashley Barnes and Steven Defour before Lingard rescued a result, backheeling Ashley Young's cross home to halve the deficit and firing in his side's second in injury time.
The result means Premier League leaders Manchester City could pull 15 points clear at the top if they beat Newcastle United on Wednesday but Lingard feels United's attitude may count for plenty come the end of the season.
"If we had played for another five or 10 minutes, we would have won the game," he told MUTV.
"[The message at half-time] was always believe. Never give up. We knew we had another 45 minutes to play so we had to be patient, keep moving the ball, moving Burnley about, and eventually we created two clear-cut chances which we scored from.
2 - Jesse Lingard is the first Man Utd substitute to score a Premier League brace since Chris Smalling did so in February 2015 (also v Burnley). Impact.— OptaJoe (@OptaJoe) December 26, 2017
"This role, further inside, allows me to be closer to the striker, to get in the box, create more things and obviously have more shots at goal. I thought we could nick it towards the end but obviously time ran out and now we look forward to the next game [against Southampton].
"We always have that fighting spirit, that never-give-up attitude and we'll see where it gets us."
Lingard's well-taken goals came after he failed to convert from point-blank range early in the second half, with goalkeeper Nick Pope blocking the ball up onto the crossbar.
"It just kind of hit me and, nine times out of 10, that would have gone in," Lingard said. "But it hit the keeper's face, bounced on to the crossbar and bounced out.
"But after that, we stayed positive and had plenty of players in the box to score goals and I'm glad that we did."