Calls to reward them for their heroics are coming from politicians along with news and social media.
Suspect Jaber al-Bakr gave elite commandos the slip over the weekend, sparking a two-day manhunt.
Police believe Mr Bakr was planning to bomb a Berlin airport.
The country's best-selling daily newspaper Bild dubbed the refugee trio the "Syrian heroes of Leipzig".
Politicians from across the spectrum recommended them for the Federal Cross of Merit, a rank in Germany's civilian honours system.
Ansgar Heveling, chairman of the parliamentary committee on interior affairs, went a step further and called on them to be granted asylum.
Fellow Syrian refugees have also taken to social media to praise their compatriots.
Police had been watching Mr Bakr, 22, for months, but he evaded police surveillance on Saturday in the eastern German city of Chemnitz, just as the authorities were preparing to storm an apartment where he was staying.
Mr Bakr, who is also Syrian, made his way south to Leipzig and asked for help and shelter from the trio, who later recognised Mr Bakr as a suspect after seeing police's appeals for information in Arabic on Facebook.
They quickly overpowered the fugitive, tied him up, and called police.
"I was furious with him, I couldn't accept something like this - especially here in Germany, the country that opened its doors to us," one of the men told RTL Television.
The three have not been identifying over fear of reprisals.
A little more than 200,000 people arrived in Germany to seek asylum in the first nine months of this year - a significant reduction from 2015, the country's interior minister Thomas de Maiziere said.
Last year, 890,000 asylum seekers arrived in the country, less than the one million originally estimated.
There is a growing clamour in Germany to honour three Syrian refugees who overpowered a bomb plot suspect with possible links to the so-called Islamic State.