The major talking point of the game came pre-match with Kurt Zouma named in the West Ham starting XI, despite an outcry over a video on social media this week where the defender was seen abusing his pet cats, only for him to pull out in the warm-up.
When the match did get underway, the visitors started well, with Jarrod Bowen racing clear before firing his side into a 10th-minute lead.
Leicester responded well and got back on level terms on the cusp of halftime through Youri Tielemans' penalty, before Ricardo Pereira completed the turnaround in the 57th minute, heading the hosts into a deserved lead.
Periera's header looked like it was going to be enough for a first win in five matches in all competitions for Leicester, only for Dawson to bundle the ball into the net from a corner, with what looked like his shoulder, to break home hearts.
The draw meant Leicester, who finished fifth in each of the last two seasons, climbed to 11th in the standings, while West Ham stayed fourth on 41 points, one above Manchester United in fifth having played one game more.
Issa Diop came in for Zouma, who was felling unwell in the warm-up, his manager David Moyes said.
Moyes was also pleased to see his side battle back and get the draw, especially given two of West Ham's rivals for the Champions League qualification places – Manchester United and Tottenham Hotspur – dropped points this weekend.
"You have to remember this (Leicester) team was winning the FA Cup and league title not so long ago," he told Sky Sports. "They have some really good players here.
"In the first half I felt annoyed after going into a good lead and gave them a leg up. It says lots for the players, we have come quickly and made massive strides in the last couple of years. I will squeeze everything out of them and I want us to challenge the top teams."
Leicester started the match poorly but battled back admirably and looked to be on course for a vital three points as they look to turn their season around, but conceding a goal from a set piece was again their undoing.
Brendan Rodgers' side have conceded 14 goals from set-pieces (excluding penalties) in the Premier League this season, more than any other team.
"We have had a few of them (late goals conceded) lately, it is deflating for the players," Rodgers told Sky Sports. "Second half we were outstanding and first half West Ham were better.
"Second half was like watching us of old, speed of the game and chances. That is seven points dropped in our last three games from good positions, it is disappointing."