Suksomskit opened the scoring in the tenth minute when Isa Halim handled the ball inside the area and the Melbourne Victory forward stepped up to score from the spot.
The visitors added a second in the 75th minute when veteran Chaiman lashed home a fine volley from Nutnum Suchao's cross before they made it 3-0 when Suksomkit was sent through on goal and, with just the goalkeeper to beat, confidently struck a low shot into the back of the net.
Singapore were awarded a penalty in the 84th minute after a player was brought down inside the area and though Mustafic scored, it was to be nothing more than a consolation.
The two sides meet again on Wednesday and Robson is keen to pick up another three points that will put Thailand in a strong position to qualify for the Asian Cup.
"It is great to win my first competitive game but my job is to qualify and I need to succeed in that and only then will I be happy," said the former Manchester United captain.
"This is a good result for us and we must now start preparing for Wednesday. It is important we recover for the game on Wednesday. It is only half the job done and we need to complete the job."
The Osasuna midfielder struck with eight minutes remaining at the Azadi Stadium in Tehran, shortly before the visitors were reduced to 10 men when Mahmoud Sahlbaeih was shown his second yellow card.
"This goal showed that we have a good leader like Javad Nekounam who can change the destiny of a match at any moment," said Iran coach Afshin Ghotbi.
"We have to accept that the time when we beat teams three or four nil has changed and teams like Jordan have improved."