The pair joined American duo Billy Hurley and Harold Varner in booking their trips to Ayrshire through the Quicken Loans National, the seventh event in The Open’s Qualifying Series.
Six others, including 2014 SSE Scottish Hydro Challenge winner Andrew Johnston, qualified for the Claret Jug joust through the European Tour’s Race to Dubai and the PGA Tour’s FedEx Cup rankings.
Sweden’s Rikard Karlberg, South Korea’s Soomin Lee and Joost Luiten from the Netherlands joined Johnston in using the Race to Dubai as their route to Royal Troon while the FedEx Cup spots were claimed by Americans William McGirt and Smylie Kaufman.
Singh, who made his first appearance in The Open at Royal Troon in 1989, will be making his 25th appearance in the event after finishing runner-up behind Hurley at Congressional Country Club in Bethesda, Maryland.
“It’s really nice to be going back to Troon again,” admitted 53-year-old Singh, whose best finish was tied for second place behind Ben Curtis in 2003 at Royal St George’s. “I love playing in The Open.”
Rahm, 21, clinched his trip to Ayrshire by finishing joint-third on his professional debut, the former amateur world No 1 having joined the paid ranks straight after claiming a share of 23rd spot in the US Open at Oakmont last Sunday.
“I really wanted to play at Royal Troon and I knew I had the exemption as an amateur but I gave it up to get my PGA Tour career started,” said Rahm. “I’m just so happy I played well today and I got my spot at Troon.
“After what I did at Oakmont, it gave me a lot of confidence on what I can do at the highest level. I have a little more knowledge of links golf now than I had before, so hopefully I can adjust fast from here and have another good run at The Open.”
Hurley, who claimed his first PGA Tour title with a three-shot hometown victory, will be making his second Open appearance at 34, having tied for 64th at at Royal Liverpool in 2014.
Varner, meanwhile, claimed his debut in the game’s oldest major at the expense of Francesco Molinari, finishing a shot ahead of the Italian.
The qualifying place that was on offer at the cancelled Greenbrier Classic on the PGA Tour will move to this week’s Barracuda Championship in Reno, Nevada.
The Greenbrier Classic was scheduled to be the last stop in the Qualifying Series, but the event has been cancelled due to the devastating flooding caused by record rainfall in West Virginia.
The scramble for players at Royal Troon continues tomorrow, when a total of 12 spots are up for grabs in Final Qualifying events at Gailes Links, Hillside, Royal Cinque Ports and Woburn.
Colin Montgomerie is among the hopefuls at Gailes Links, where the eight-time European No 1 heads out at 8.10am in his first round in the company of fellow Scot Jack McDonald and Frenchman Romain Wattel, who won the Scottish Open Stroke-Play Championship on the same course in 2010.
Scottish duo Janie Moodie and Dean Robertson steered Europe to en emphatic 18.5-11.5 victory over the United States in the Palmer Cup at Formby.
Unlike two years ago, when Grant Forrest, Jack McDonald and James Ross were on a winning team at Walton Heath, the home side didn’t have a Scot in its ranks on this occasion.
But Moodie, a two-time LPGA winner, and former Italian Open champion Robertson played their part in seeing another amateur trophy join the Walker and Curtis Cup in now residing on this side of the Atlantic as head coach and assistant coach respectively.
“I think the secret was having 10 guys who were mentally prepared and a few guys – Adrian (Meronk) for example – who wanted revenge for losing last year,” said a delighted Moodie said. “They were all really focused.”
Stirling University player Mathias Eggenberger beat former US Junior champion Will Zalatoris by one hole as Europe won the final singles session 5.5-4.5.
The United States now holds a 10-9 lead in the series, with one match halved.
Ayrshireman Euan Walker finished with a flourish to record an impressive six-shot victory in the East of Scotland Open at Lundin.
The Kilmarnock (Barassie) player closed with a 64 - the best effort of the weekend - for a 13-under-par 271 total as he claimed a second Scottish Order of Merit title triumph.
Walker stormed home in 28, which included an eagle-2 at the 16th as well as birdies at the 10th, 11th, 13th, 14th and 17th.
Maintaining his eye-catching run of results, Turnhouse man Euan McIntosh finished second on 277, with Adam Fisher (Newmachar) and Duncan McNeill (Powfoot) two shots further back in third.