An impressive goal from midfielder Yaser Kasim put Iraq in front with 13 minutes remaining, as he slalomed past three opponents before his shot took a deflection to beat Jordan goalkeeper Amer Shafi.
It continued an improbably impressive Asian Cup for Swindon Town with two of the League One club's midfielders having played key roles in the opening four days of the continental championships.
While Kasim, who was born in Baghdad but moved to England as a seven-year-old, struck the winner for Iraq, his club-mate Massimo Luongo was man of the match for host nation Australia in their 4-1 triumph over Kuwait on Friday.
With both teams expected to lose to Japan and beat Palestine in their other Group D fixtures, Jordan - who finised the game with 10 men - and Iraq were clearly wary of giving their main rival for a quarter-final berth too much leeway.
A rather poor first half included just two shots on target as both teams struggled for cohesion in their respective forward thirds.
Iraq skipper Younis Mahmoud was busy throughout the opening period and had a big chance to break the deadlock in the 16th minute but, after a clever one-two with Kasim, the veteran striker's shot was deflected wide by Tareq Zeyad.
Iraq looked better up until half-time with Kasim, Alaa Abdul-Zahra and Amjad Kalaf linking well in midfield, but the final pass was lacking.
By contrast, Ray Wilkins' Jordan struggled to gain possession and generally resorted to long crosses into the box that Iraq goalkeeper Jalal Hassan Hachem comfortably dealt with.
Iraq had a penalty shout turned down in the 55th minute when Dhurgham Ismail went down acrobatically, although replays suggested he had a case.
On the hour mark, Jordan almost scored twice with Iraq defender Ahmad Ibrahim looping a header just over his own crossbar before Mohammed Mustafa's shot from the corner went straight at Hachem.
But just as Wilkins' men appeared to be improving, Kasim struck in the 77th minute, while Jordan defender Anas Bani Yaseen was sent off seven minutes later.
Abdul-Zahra should have doubled Iraq's lead in the closing stages but sent a powerful header straight at Shafi, although it did not matter as the 2007 Asian Cup champions held on to win and move second in Group D, behind Japan on goal difference.
