Jermain Beckford's 89th minute header looked to have earned Jamaica a point after the United States had led through a Jozy Altidore goal on the half hour.
But Michael Bradley slipped the ball inside the area to Evans who, with a surprising amount of space, drove past Jamaican keeper Donovan Ricketts to seal the win.
The victory moves the United States to seven points from four games as they bid to earn one of three automatic qualification spots from the six-team CONCACAF final qualifying group.
Costa Rica, who lead on goal difference, and Mexico are also on seven points although the Mexicans have played one more game.
Panama are on six points, Honduras have four and Jamaica's chances now look slim as they sit at the bottom of the table with just two points.
It wasn't an entirely convincing performance from the U.S. but coach Juergen Klinsmann was pleased with the character his team showed as they responded to the blow of a late equalizer to get their first ever win in Kingston in a qualifier.
"They were angry with themselves about allowing that goal and they responded," said the former German national team striker and coach.
Klinsmann's team started brightly and went close to an opener in the second minute when Bradley struck the post with a drive from 20 yards out.
The United States took the lead when Altidore, who had an early effort ruled out for offside, rose to power in a superb cross from winger Graham Zusi.
Nine minutes later Jamaica almost drew level when Rodolph Austin burst through on to a Ryan Johnson pass but his low shot struck the post.
The United States looked in charge but with a minute of regulation time remaining, Beckford was left unmarked to head home from an Austin cross.
However, there was more drama to come and three minutes later the United States restored their advantage to get their first away win in qualifying.
There were some other major positives for Klinsmann as Evans looked at ease in a new role at right-back, Zusi played with intelligence on the right flank and Altidore, who scored in Sunday's 4-3 friendly win over Germany, looked to be full of confidence.
The United States host Panama in Seattle on Tuesday and finish their trio of June qualifiers against Honduras in Salt Lake on June 18.
(Reporting By Simon Evans; Editing by Peter Rutherford and Amlan Chakraborty)