Greece qualified for a second successive World Cup after another Kostas Mitroglou goal secured a 1-1 draw with Romania in the second leg of their playoff on Tuesday and a 4-2 victory on aggregate.
Mitroglou, who scored twice in Greece's 3-1 first-leg win, beat the offside trap midway through the first half and gave keeper Ciprian Tatarusanu no chance with his 22nd goal in all competitions this season.
Romania equalised 10 minutes after the break when the ball ricocheted off defender Vasilis Torosidis from Jose Holebas's attempted clearance and looped beyond goalkeeper Orestis Karnezis for an astonishing own goal.
Greece, the 2004 European champions, competed in the previous World Cup finals, in South Africa in 2010, as well as the 1994 tournament in the United States, but they have failed to get past the first round, losing five of their six matches.
Greece arrived in Bucharest after seven consecutive wins and aiming to produce another solid defensive display, with Bogdan Stancu's strike at the Georgios Karaiskakis Stadium on Friday ending their run of 594 minutes without conceding a goal.
"It's a great moment for all of us and for all Greeks," Fulham midfielder Karagounis, who assisted Mitroglou with clever through-ball for the opening goal after 23 minutes, told reporters.
"I've been playing for the national team for 14 years, I played at the European championship's finals 10 years ago and now I fulfilled another dream."
The home side, who have failed to reach the World Cup finals since 1998, dominated the early stages with winger Gabriel Torje continually tormenting the Greece defence on the right flank but were unable to create any clear-cut chances.
It was Greece who almost broke the deadlock after 17 minutes - in their first attack of the game - when Ciprian Tatarusanu produced a brilliant save to deny Holebas's fierce left-foot effort.
Tatarusanu and centre back Vlad Chiriches returned to Victor Piturca's starting line-up after recovering from injuries and midfielders Ovidiu Hoban and Alexandru Maxim replaced Alexandru Borceanu and Razvan Cocis.
"This qualification was important for me and the players, but it's important for the Greek people too," said Santos.
"We knew that Romania will give everything possible but we're well-prepared. We managed to go 1-0 up and entered the second half with a good mood.
"Then we conceded a strange goal but we reacted well we didn't allow them having a lot of possession and creating scoring opportunities.
"We're confident of our qualification. We have lost only three times in (the last) 26 official matches."
More than 50 000 Romanian fans created an intimidating atmosphere for the visiting players but Piturca's side looked bereft of ideas, and strikers Ciprian Marica and Stancu struggled against the well-organised visitors.
After the bizarre own goal, the hosts upped the pressure with Hoban forcing Karnezis make his first real save with a low shot from the edge of the area on the hour-mark but were unable to force their opponents making mistakes.
The Romanians failed to capitalise on the Greek defensive midfielders' lack of pace as they went for a number of crosses which caused little problems for tall Dimitris Siovas and Sokratis Papastathopoulos.
Greece, who went into the playoffs after finishing behind European qualifying Group G winners Bosnia on goal difference, held their nerve, passing the ball around confidently for much of the second half.
Romania's failure - they were also knocked out of another World Cup playoff in 2001, to Slovenia - was the latest setback for Piturca as the 57-year-old was also in charge of the Romanian teams that missed the 2006 and 2010 finals.
"Unfortunately for us, Greece were the better team in both games and we have to congratulate them," said Piturca.