Blues manager David Jeffrey had admitted concerns over the heat's impact on his players in the build-up to the second qualifying round first leg.
However, the Northern Irish side faced similar energy-sapping conditions against AEL Limassol in last season's Champions League and had clearly learned from that experience.
Burns grabbed the winner in just the 25th minute, firing past Christian Gimenez after good work down the left.
Xanthi enjoyed the lion's share of possession after the goal, but Linfield held on to take a 1-0 lead into next week's second leg at Windsor Park.
There was another shock result in Scandinavia, where Frazer Wright secured St Johnstone a 1-0 victory against 22-time Norwegian champions Rosenborg.
The centre-half made the most of a brief period of Saints pressure in the opening stages to net from close range and secure his side a precious away goal.
It was quite a start to manager Tommy Wright's reign, but there were worrying moments following the 19th-minute strike and they were forced to defend stoutly to protect their lead.
Derry were another side to impress in Europe as they gave Trabzonspor a scare.
Adrian Mierzejewski and Paulo Henrique efforts gave the Turkish side a two-goal lead within 16 minutes, only for City to come storming back.
David McDaid pulled one back at the Huseyin Avni Aker Stadium, before Patrick Kavanagh drew parity in the 32nd minute.
It had looked like another shock could be on the cards, only for an own goal from captain Barry Molloy to put Derry on the back foot before half-time.
Giray Kacar put Trabzonspor further ahead early in the second half but City continued to threaten, with Ryan McBride firing just wide in injury time as the match finished 4-2 to the hosts.
There was also defeat for Hibernian, who face an uphill battle in next week's second leg tie with Malmo.
The Swedish league leaders secured a comfortable 2-0 win this evening thanks to first-half goals from skipper Jiloan Hamad and Magnus Eriksson.
Prestatyn Town's task next week looks nigh on impossible after suffering a 5-0 hammering in Croatia.
Rijeka came into the match as overwhelming favourites and did not disappoint, with former international Leon Benko netting a hat-trick before Vedran Jugovic and Damir Zlomislic added gloss to the scoreline.
- Sports & Recreation
- AEL Limassol