A superb strike from Michael Bradley gave the US a bonus point in the quest to reach next year’s World Cup, and a big psychological boost.

A squad still finding their feet under Bruce Arena shrugged off a dismal history at the Estadio Azteca to claim a draw on Sunday night, looking liberated by the lack of pressure that comes with limited expectations.

The captain, whose attacking prowess has not been to the fore in recent matches, intercepted the ball in midfield in the sixth minute, dashed forward and, spotting the goalkeeper out of position, chipped him from 40 yards for a sensational opening goal. The shot sank over Guillermo Ochoa in a mesmerising arc: a piece of craftsmanship, imagination and beauty so perfect that a video replay should run on an endless loop in a permanent exhibition at the Metropolitan Museum of Art.

“Obviously it gave us an advantage early on,” Bradley told Fox Sports 1. “They had a lot of the ball but tactically the game went the way we expected … any time you get a point here it’s great, the mentality and the commitment from every guy was really good.” Bradley almost scored again from distance inside the last 20 minutes with a thunderbolt that clipped the outside of a post.

Carlos Vela scored Mexico’s goal, but a point was most welcome and fully merited for the US after an obdurate display in Mexico City against the side that leads the qualification standings. The US moved level on points with Costa Rica, who are second, although the Americans have played a game more.

Since Mexico won the previous meeting between the teams, the outcome also stands as a yardstick indicating improvements since Arena became coach. “We had let a lot of little things drop and I think Bruce has come in and done a really good job,” Bradley said in his post-match interview.

Mexico beat Honduras 3-0 last Thursday and - in stark contrast to their turmoil four years ago - have looked impressive in this qualifying campaign under their coach, Juan Carlos Osorio, a Colombian who is a former coach of the Chicago Fire and New York Red Bulls and a one-time student in Liverpool.

They also looked good a year ago in the Copa America Centenario before the bizarre blip in the quarter-finals when they lost 7-0 to eventual winners Chile. Here they were missing the injured Rafael Marquez, Miguel Layun and Andres Guardado, though Carlos Salcedo made an unexpected recovery from a leg injury.

Starting off the final Hexagonal round of Concacaf qualification, Mexico beat the US 2-1 in Columbus, Ohio, last November thanks to a late Marquez goal. It was an occasion freighted with political as well as sporting import, three days after the election of a certain former failed sports franchise owner to the White House.

The outcome ended the Americans’ 30-game home unbeaten run in World Cup qualifiers as well as the “dos a cero” scoreline streak. When the US lost to Costa Rica a couple of days later, Jurgen Klinsmann’s tenure as head coach was doomed.

Arena had said after the win over Trinidad & Tobago last week that he would change the team; he transformed it, with seven new names in the starting XI. Perhaps most surprisingly, the in-form Tim Howard, who played in the Americans’ only win at the Azteca, a 1-0 friendly victory in 2012, was rested and replaced in goal by Brad Guzan.

Geoff Cameron, Bradley, DeAndre Yedlin and Christian Pulisic were the only four to retain their places; left back Jorge Villafana was injured. The energetic 21-year-old, Kellyn Acosta, of FC Dallas, came in to act as a midfield disrupter in a formation that was a 3-4-3 in possession. It is the sort of dramatic rotation that, were Klinsmann still in the dug-out, might have inspired insinuations about his competence and/or sanity. But Arena’s pedigree, demeanour and the current performances of the team inspire more trust in his judgment.

The short rest between games was undoubtedly one factor; another, perhaps, was that with three points collected against T&T, Arena could afford to experiment at a venue where avoiding defeat felt likely to be a long-shot no matter who he picked.

While Arena understandably fretted about the potential physical effects of the Azteca’s high altitude its factor as a competitive advantage risked being exaggerated; only two of the Mexico line-up play their club football in the country. The pace of the game hardly dipped as the night wore on, with both teams seeking a winner.

The 18-year-old Pulisic has become the US’s get-out-of-mediocrity free card, as he was in the fitful 2-0 win over bottom side T&T in the Denver suburbs last Thursday, scoring two second-half goals. On Sunday, though quieter, he made an adroit run and pass to set up a chance for Yedlin late in the first half, and an audacious attempted sand-wedge of a pass over the back line that was cleared.

Pulisic had boldly predicted victory ahead of this game. He had the opportunity to make his prophecy come true with two minutes remaining as he fashioned an enticing chance, but shot wide from the edge of the area. This time, for once, he would not be the headline-maker.

It was the kind of intense occasion that would have benefitted from firm officiating. But in the first three minutes, Salcedo was lucky to escape a booking for an elbow on DaMarcus Beasley, and seconds later, ought to have been sent off for another flailing arm that caught Bobby Wood.

Then came Bradley: it was the Toronto FC midfielder’s second international goal since July 2015 and a rare moment of adversity under Osorio; only the second goal Mexico had conceded in six Hex matches.

Chicharito should have levelled after 20 minutes when set clear, but with the ball bouncing, mishit his lob over Guzan and injured himself colliding with the goalkeeper, though he played on. The contest was tied soon after. On the counter-attack, Beasley – in his US-record fifth cycle of World Cup qualifiers – was unable to stop Vela cutting inside and dribbling along the edge of the penalty area, giving him enough of an angle to lash a low shot beyond Guzan. The US defender Omar Gonzalez fluffed a free header inside Mexico’s box as both teams looked dangerous, but went in at the break locked at 1-1.

More direct than Mexico – who worked the ball from side to side in search of space that the visitors mostly closed down effectively – the US were spirited in the second half and created a couple of half-chances.

Still, Mexico almost scored again with 20 minutes left as Hector Herrera cannoned a free kick off Guzan’s crossbar. Then came Bradley’s effort that hit the post. As rain fell in Mexico City, the Americans held on amid late pressure.

With a goalscoring turn from one of his stars and solid displays from lesser lights, Arena’s choice of players was vindicated and suggests a degree of depth that will give him the kind of selection problems that all managers relish as the US heads into this summer’s Gold Cup feeling good about making it to Russia.