The result leaves Montenegro on 14 points and England on 12, with four qualifying rounds remaining — including the return fixture at Wembley in October. Poland is third with eight points.
England took the lead in the sixth minute when Wayne Rooney headed in a corner from Steven Gerrard after being left unmarked in the area.
Second half substitute Damjanovic levelled from close range in the 76th minute after a melee in a packed England goalmouth, securing a third consecutive draw between the two teams in international qualifiers.
England dominated most of the first half, creating several good chances, while Montenegro failed to penetrate a well-organized defence. However, the hosts took control for all but the latter stages of the second half, securing a vital result in Montenegro's qualifying campaign.
"To win this game we had to play in second half like we did in the first," England coach Roy Hodgson said. "I thought we passed the ball well and had a lot of control in the midfield, but in the second half we unfortunately lost that control and Montenegro had the good spell and took the game to us."
"We were playing with fear in the first half, but then we showed our strengths," Montenegro striker Mirko Vucinic said. "This is a great result for us, that gives us a good chance to qualify for the World Cup."
Rooney's goal capped an early England offensive in which the Manchester United striker, who was sent off in a Euro 2012 qualifier in in Podgorica, hit the post with a spectacular lob from the right side of the area in the third minute. Glen Johnson then grazed the bar in the sixth with a fierce shot from some 30 metres.
Gerrard and James Milner meanwhile controlled the midfield, leading several dangerous attacks in the opening stages.
Rooney just failed to convert Milner's driven pass across the face of the goal in the 27th minute, while striker Danny Welbeck had a penalty appeal turned down — and instead received a yellow card for diving.
With nothing to lose, Montenegro started attacking right from the start of the second half.
"I told my players at halftime that they have nothing to fear from the England team," Montenegro coach Branko Brnovic said. "I told them they are not better than us. And, they delivered."
Under pressure, the England players went from retaining the ball to being completely overrun and failing to create any real attacking threat.
"I don't think this is a particularly bad result because Montenegro is a difficult team to beat," Hodgson said. "Montenegro and Poland will have to come to us and we think it is still in our hands and we intend to make sure by winning these matches."
The first chance for Montenegro came in the 47th minute when Stevan Jovetic's header narrowly missed the target. Ten minutes later, the Fiorentina midfielder tried to lob England goalkeeper Joe Hart with a long-range effort on the break.
Vucinic unleashed a shot in the 72nd that fizzed past the post, and minutes later Hart needed to make a diving save to deflect Jovetic's drive. But there was nothing he could do when Damjanovic rifled the ball home.
England reacted by pushing forward in search of a winner and a free kick from Gerrard drew a fine save by Montenegro goalkeeper Mladen Bozovic in the closing stages.
The draw will be a setback for England coach Roy Hodgson, who had made several changes from the team that thrashed San Marino 8-0 on Friday — bringing in Gerrard for his 102nd cap, along with fellow midfielders Michael Carrick and James Milner, plus Ashley Cole and Welbeck.
Montenegro went into the game spurred by an eight-match unbeaten run, including a 1-0 victory in Moldova on Friday.
The tiny mountainous country of just 625,000 people has established its credentials as a footballing nation after splitting from Serbia in 2006. Its first official international competition as an independent nation was in 2010 World Cup qualifying.