The USA had come from behind to lead 2-1 on Sunday and looked to have ended Portugal's FIFA World Cup campaign in Manaus, before Cristiano Ronalo got free on the right in the fifth minute of second-half stoppage time and picked out Varela, who powered a header past Tim Howard.
An early goal to Nani saw Portugal respond well to their devastating 4-0 loss to Germany last week, but second-half strikes to Jermaine Jones and captain Clint Dempsey had put the USA in front and on the verge of securing a berth in the round of 16 for the second tournament in a row.
But Varela's late header ensured the top two positions in Group G remain up for grabs with Germany (four points) on top, level with the USA, while Ghana and Portugal both have slim chances of advancing with one point each.
Despite speculation that a knee injury would keep him out of Portugal's line-up, Ronaldo started, while Helder Postiga, Ricardo Costa, Andre Almeida and Beto replaced the unavailable quartet of Hugo Almeida (thigh), Pepe (suspension), Fabio Coentrao (groin) and Rui Patricio (thigh).
Klinsmann picked Graham Zusi to replace the injured Jozy Altidore (hamstring) with the Sporting Kansas City midfielder lining up on the left and Dempsey moving into a lone striker's role.
The match started horribly for the USA with Nani opening the scoring in just the fifth minute.
Miguel Veloso's poor cross bounced in front of Geoff Cameron and the central defender sliced his clearance to the back post, where Nani took a touch and then smashed his shot high into the net with Howard stranded.
Portugal pinned the Americans back in the opening stages with Klinsmann's side creating their first decent opportunity when Fabian Johnson, who was a regular outlet pushing up from right-back, got free in the 10th minute but his shot was deflected for a corner.
Six minutes later, Portugal's injury curse seemingly struck again with Postiga getting subbed off for Eder with an apparent hamstring strain.
The substitution seemed to halt Portugal's momentum, with Michael Bradley and Dempsey each failing to convert a pair of half chances for the USA before the half-hour mark.
Portugal reminded the USA of their counter-attacking threat three minutes before half-time with Nani unleashing a scorching drive that was parried by Howard, while in the 45th minute, the American gloveman made a double save to ensure his team were just 1-0 behind at the break.
Although Howard was unimpressed by his initial save onto the post from a strike from Nani, the 35-year-old did brilliantly to keep his balance and tip Eder's follow-up over the bar.
Almeida came off at half-time with another possible injury, with Veloso moving to left-back and William Carvalho promoted to defensive midfield.
The second half almost started exactly as the first had with Eder acrobatically volleying over the bar after another poor clearance from Cameron, before Johnson got forward to force Beto into a save.
In the 55th minute, Bradley should have equalised when Johnson again got free on the right, cutting it back but the Toronto FC midfielder's shot was blocked on the line by Costa.
The USA notched a deserved equaliser in the 63rd minute when a corner was cleared to Jones and the Schalke midfielder smashed his shot inside the far post, while Howard made another sharp save soon after to deny Raul Meireles.
Although Portugal had a few chances to lead again, the USA hit the front with nine minutes remaining when Zusi collected a loose ball in box and crossed for Dempsey to score off his midriff, although Klinsmann's men were denied victory thanks to Varela's late heroics.
