Australian Day made five birdies and managed to keep the bogeys off his card as he began his campaign at the first Major Championship of the year in favourable conditions on Thursday morning.
That gave him the early clubhouse lead on five under par and, with the wind picking up in the afternoon at TPC Harding Park, it looked like the later starters would be unable to catch him.
But American Todd, who held the 54 hole lead at last week's WGC-FedEx St Jude Invitational, got three birdies on his back nine before pulling off a sensational par save at the ninth - his last - to stay alongside Day at the summit.
Koepka made six birdies and two bogeys while German Kaymer carded an eagle, four birdies and two bogeys - the last of which came at his final hole to deny him a share of the lead.
Englishman Justin Rose, France's Mike Lorenzo-Vera and Americans Scottie Scheffler, Xander Schauffele, Bud Cauley, Zach Johnson and Brendan Steele were also a single shot behind Day and Todd.
Day started at the tenth tee and picked up a shot at the earliest opportunity, sending his approach into the par five to three feet before rolling in for a birdie.
After a run of six straight pars, Day got another close range birdie at the short 17th to reach the turn in 33.
Day made further gains at the first and fourth before finishing with a birdie on the ninth - his last - thanks to a wonderful second shot which bounced to within six feet of the cup.
He was happy with his first round effort. He said: "It was good. I got off to a great start. Got a little bit lucky between the bunkers on ten, but you've got to take the luck when it comes by.
"Today I drove it really nicely and when I was out of position I left myself on the right side of the fairways to be able to at least get somewhere around the greens, and if I did miss the greens I left it in the right spot.
"Yeah, pretty sound the whole way around. I hit a lot of good quality iron shots coming into the greens. Wasn't overly aggressive. And there were a couple of shots on nine and 17 where I hit it pretty tight, but overall it was very solid."
Starting from the tenth in the afternoon, Todd reached the turn in 33 after mixing four birdies with two bogeys on his front nine.
He tapped in from close range for another gain at the first following a lovely approach shot, before holing lengthy birdie putts at the fifth and seventh to join Day at the top of the leaderboard.
After knocking his birdie putt ten feet past the hole at the ninth, he held his nerve to save par and remain alongside Day.
He said: "The front nine I think has more birdie chances, and I drove it really well on that nine.
"I was struggling off the tee on the first nine, so I felt like I did a great job just scraping it around and giving myself a chance to shoot a good nine coming in."
Kaymer, who came into this event having played just once since March due to the coronavirus hiatus, dropped an early shot at the 11th but made birdies at the 14th and 17th to turn in 34.
The Ryder Cup star then followed up birdies at the first and third with a superb eagle from over 45 feet on the long fourth to jump into contention.
He then made a miraculous up-and-down on the fifth, holing his par putt from over 20 feet having been in the run off area after two shots.
That saw him retain his share of the lead but he narrowly missed his par putt on the ninth - his last - to fall back to four under.
Kaymer, who watched a replay of the back nine of the final round of his 2014 U.S. Open win at Pinehurst as he looked for inspiration on Wednesday evening, was delighted with his long range eagle.
He said: "That eagle was very special. I hit a good tee shot and then I thought I needed a bullet with a three wood to get to the green.
"I thought if I maybe placed it in the right bunker I'd have a good chance to make birdie and then I pulled off that bullet and it got onto the green, but it was still quite far away, so you're thinking maybe a two putt birdie would be great, then it went in."
Koepka, who is aiming for a historic hat-trick of wins at this event, failed to take advantage of the par five tenth - his first - and dropped a shot on the next, but then birdied the 13th, 14th, 16th and 18th thanks to some excellent approach play.
A bogey on the first slowed Koepka’s momentum, although he did well to limit the damage to a single stroke after only moving his second shot 40 yards from the rough and then finding a bunker with his third.
He bounced back with birdies on the second and fourth and was pleased with his opening effort.
He said: "I feel good, I feel confident. I'm excited for the next three days. I think I can definitely play a lot better and just need to tidy a few things up and we'll be there come Sunday on the back nine."