Referree Barry Knight was the centre of attention throughout as he sent off a player on both sides - awarding two hotly disputed penalties in the process.
But unlike Bournemouth, the visitors were unable to score from the penalty spot as they slumped to yet another away defeat.
Swindon were reduced to 10 men in the 24th minute when Andy Gurney was adjudged to have fouled on-loan striker James Keene in the box.
Bournemouth striker Hayter stepped up to convert the spot-kick for his third goal of the season.
And last season's top scorer for the Cherries was close to adding the second 14 minutes later when he just failed to connect with Steven Foley's cross.
But the home side suffered a setback one minute before the break when 10-man Swindon got back on level terms.
The Bournemouth defence failed to cut out Michael Pook's long throw and Steve Jenkins ghosted in at the far post to head Swindon level.
Bournemouth pushed after the break for another goal and were eventually rewarded on 68 minutes when Hayter headed in his second from James O'Connor's precise cross.
Swindon then blew a golden opportunity to level when referee Knight then awarded them a penalty seconds later.
Striker Tony Thorpe ran on to Gareth Whalley's through ball only to be hauled back by defender Josh Gowling in the box.
Much to the amazement of the home fans, referee Knight then awarded a penalty and sent off Gowling in the process.
But the home fans' dismay was short-lived when Stewart produced a wonder-save to keep out Thorpe's penalty.
And it was a double joy for the keeper when the offside flag went up after Thorpe banged in the rebound from Whalley's centre.