The 22-year-old joined Dungannon five years ago after impressing for County Tyrone in the Milk Cup tournament.
"Despite being offered a new contact by the club Jamie has opted to sign for Crusaders," said the Swifts.
Crusaders will start the 2017/18 campaign with a Europa League qualifier first leg against FK Liepaja of Latvia at Seaview on Thursday night.
"All the best in the future Jamie and thanks for your commitment during your five seasons at Stangmore Park," added Dungannon on Tuesday.
Glackin's arrival at the north Belfast club follows the close-season signings of Danish goalkeeper Brian Jensen, defender/midfielder Sean Ward and striker Mark McAllister.
Crusaders missed out on a third straight Premiership title last season and boss Stephen Baxter will be determined to regain the Gibson Cup.

It will operate five times weekly until 12 June, daily from 13 June to 2 September and four times weekly between 3 September and 6 October.
The first flight was inaugurated at a ceremony at Edinburgh Airport.
The airport said it marked "a significant step" in achieving better connectivity between the Scottish capital and North America.
Airport managing director Gordon Dewar added: "This new service will give Scottish passengers the opportunity to visit one of America's most exciting and vibrant cities and take full advantage of the onward travel options to many other US destinations.
"We're delighted to continue our good relationship with United and celebrate this new route's inaugural flight."
Deputy First Minister Nicola Sturgeon said the service highlighted the strong relationship between Scotland and the United States.
"This service will provide real benefits to Scotland's economy, encouraging more tourism, trade and investment by reducing the need for travellers to make connecting flights via London or the continent," she added.

The Japanese carmaker said it had suspended its global and Japanese sites early on Wednesday after they had been bombarded with traffic.
Hacktivists linked to the Anonymous collective have claimed responsibility. They tweeted the attack was "punishment" for Japan's killing of whales and dolphins.
Nissan has said that it has no view on such hunting activities.
Over recent weeks, Anonymous has struck against the websites of the Japanese president and other government departments in protest against the animals' deaths.
This follows related cyber-attacks against Icelandic institutions in November as part of the same OpWhales campaign, which resulted in most of the country's government sites being made unavailable for about 13 hours.
One of the hackers claiming responsibility for the attacks explained why Nissan had been targeted.
"They are a big corporation in Japan, and we have targeted big corporations to spread awareness about the killing [of dolphins] in the cove in Taiji because the Japanese news is censoring it," he said.
"As a note for Nissan, we are not out to harm your customer data or system data."
The attack coincides with the Detroit Auto Show, where Nissan unveiled a concept pickup van on Tuesday.
The company's US and European websites remain online.
"At Nissan, customer privacy and security is of utmost importance, and we take any potential threat to our information systems seriously," said a spokesman.
"Because of a potential distributed denial of service [DDoS] attack, we are temporarily suspending service on our websites to prevent further risks.
"Nissan continuously monitors and takes aggressive steps to ensure the protection of our information systems and all of our data."
Crusaders have snapped up midfielder Jamie Glackin from Dungannon Swifts on a two-year contract.
[NEXT_CONCEPT]
The first direct flight service between Edinburgh and Chicago has been launched by United Airlines.
[NEXT_CONCEPT]
Two of Nissan's main websites have been driven offline by a cyber-attack.