Lingard converted a rebound from his own saved penalty in the 21st minute for his fourth goal in six appearances since arriving on loan from Manchester United.
Dawson headed in West Ham's second goal seven minutes later to help his side up to fifth spot with 48 points from 27 games.
David Moyes' side are two points behind fourth-placed Chelsea having played a game less.
"We were resilient, [but] we didn't play well tonight," Moyes, who helped West Ham avoid relegation last season after replacing Manuel Pellegrini, said. "Not a good performance by our standards. The players are disappointed.
"Leeds United do a lot to you and make it difficult to play. We had a bit of control but I didn't enjoy it."
West Ham's best finish since returning to the top flight in 2012 is seventh in 2016, while the last time they finished in the top four in the top flight was in 1986.
"When you get near the top it gives you a big incentive," Moyes said. "I was gutted when we dropped out the top four.
"We are doing really well and having a good season. It could be a great season. I'm not saying we will finish (in the top four) but we need to hang in and see what we can do."
Leeds twice had goals ruled out before West Ham scored and squandered several good opportunities to get back into the game.
In the end a frustrating night for Marcelo Bielsa's side left them in 11th spot.