Mathews came out to bat in the 25th over of the Sri Lanka innings but his helmet chinstrap broke just as he was preparing to face bowler Shakib Al Hasan.
Mathews gestured to the dressing room and was waiting for a replacement helmet while Shakib was seen chatting with the umpire as the clock ticked.
Under the tournament's playing conditions a new batter should be ready to face the ball within two minutes of a dismissal and, following an appeal by Shakib appeal, umpire Marais Erasmus ruled Mathews had been timed out.
Mathews remonstrated with the umpires and threw his helmet in disgust soon after crossing the boundary rope.
After Bangladesh won the match by three wickets, eliminating their opponents in the process, Sri Lanka's players refused to shake hands with their counterparts.
Mathews later launched a broadside against Bangladesh and the match officials for punishing him for what he said was a simple case of "equipment malfunction".
"Obviously disgraceful from Shakib and Bangladesh," a furious Mathews told reporters at the Arun Jaitley Stadium.
"If they want to play cricket like that, stoop to that level, obviously there's something wrong drastically..."
"It's (lack of) common sense, and bringing the game into disrepute. Absolutely disgraceful."
The 36-year-old said he was ready to take strike within the stipulated two minutes but the broken chinstrap meant he could not risk his safety by facing a bowler without a proper helmet.
"In my 15-year-old career I've never seen a team going down to that level," he said, promising to furnish video proof of being ready to take guard within two minutes before his chinstrap broke.
Mathews said the on-field match officials should have checked with the fourth umpire before declaring him out but maintained Bangladesh should have recalled him even after their appeal.
"I don't think any other team would do that because it was black and white. It was equipment malfunction, the helmet coming off. It was a big safety issue as well."
"Shakib had the option (of recalling) but he decided to go the other way."
Addressing the press conference before Mathews, Shakib, whose all-round performance earned him the player-of-the-match award, was equally belligerent about the episode.
Asked if he regretted making the appeal, Shakib said, "Not at all.
"One of our fielders came to me and said, 'If you appeal, the law says he's out because he hasn't taken guard within the timeframe'."
Shakib said he was asked by the umpire if he wanted to recall Mathews after the appeal had been upheld.
"I said he's out, if you call him back, it doesn't look good. So I said I won't call him back ... yes unfortunate but within the rules."
Asked if he thought it violated the spirit of the game, Shakib shot back by saying "Then (the) ICC (International Cricket Council) should look into it and change the rules."
The 36-year-old was equally combative when asked what he would have done had he been in Mathews' shoes and said, "I'll be careful that it doesn't happen to me."
The all-rounder said had known Mathews well since their under-19 World Cup days and that the Sri Lankan approached him after being given out.
"He came and asked me whether I'd withdraw my appeal ... I said 'I understand your situation. It's unfortunate, but I don't want to'.
"We had to win this match. When you are fighting a war for your team or country I believe everything is fair."
There have been at least half a dozen cases ofa player being timed out in first-class cricket, the most recent being Zimbabwean Charles Kunje's dismissal in a Logan Cup match in 2017.
