USA were unable to make the most of a number of golden opportunities in a dominant first half performance until Pulisic gave his side a deserved lead in the 38th minute, with an effort that ultimately cut short his evening at half time after colliding with Iran goalkeeper Alireza Beiranvand.
In contrast to the first half, the Stars and Stripes had to remain resilient in the second period as Iran pressed forward in search of an elusive equaliser, and Gregg Berhalter's side also survived a late penalty appeal to secure their passage to the knockout rounds.
Iran went into the crucial final group game knowing a draw would likely be enough to progress to the knockout stages, while USA were fully aware that only a win would do if they were to continue their journey in Qatar.
Both sides enjoyed moments of possession in the opening stages, but the first real shot in anger came from the right boot of Yunus Musah, who fired the ball over the crossbar.
After failing to hit the target, Musah turned his attention to the creative aspect of his game, and his precise lofted cross was headed into the grateful clutches of Beiranvand by Pulisic.
USA continued to look the more threatening of the two sides in the first half, but they were struggling to find a cutting edge, as Timothy Weah spurned a couple of chances.
The forward guided a tame header straight at Beiranvand, before thumping another promising opportunity into the stands.
Their endeavour and persistence was finally rewarded in the 38th minute when Sergino Dest calmly nodded Weston Mckennie's ball into the path of Pulisic, who collided with Beiranvand in the process of scoring the opening goal, and the Chelsea winger was subsequently forced off the break.
Moments before half time, Weah thought he had doubled his country's lead when he latched onto Tyler Adams's piercing through ball and slotted a neat shot beyond the goalkeeper, however, the assistant's flag cut the celebrations short.
Despite offering very little in the opening 45, Iran were given a boost when England scored two early goals in the second half against Wales, which effectively meant a point would be enough for Team Melli to progress.
Saman Ghoddos was introduced into the fray at the break, and he would have been relieved to see the referee award a foul in the build-up to his wasteful effort from inside the box, which flew past Matt Turner's far post.
Despite desperately needing to find a leveller, Iran found it hard to create clear-cut chances against a resilient USA defence that were determined to keep consecutive World Cup clean sheets for the first time since 1930.
With the deficit still a narrow one, USA began to protect their slender advantage, and as they dropped deeper, Iran began to offer an improved threat in the final third.
Ali Karimi was unable to make meaningful contact from a teasing delivery from the right flank, while Morteza Pouraliganji powered a diving header agonisingly wide in the nine minutes of additional time that were played.
In the final moments of the contest, Iran saw their vociferous penalty appeals turned down for what they felt was a foul on Mehdi Taremi by Cameron Carter-Vickers, but referee Mateu Lahoz and VAR deemed the incident not worthy of spot-kick, ensuring USA held on to claim three crucial points
While Iran's 44-year wait for an appearance in the World Cup knockout rounds will continue, USA will prepare for their last 16 game against the Netherlands, which will take place at the Khalifa International Stadium on Saturday.