"We won the match, that's what's important," striker Zlatan Ibrahimovic told TV4.
The Swedish side created a number of scoring chances early in the match, and seemed ready to dominate.
At the 30 minute mark, defender Mikael Lustig put Sweden on the scoreboard with a shot off a pass from Kim Källström that deflected off a Moldavian player and into the goal.
It was Lustig's first goal for Sweden in international play.
Sweden's intensity waned following the goal, however.
"It was like we became a little hesitant," coach Erik Hamrén told TV4.
"We have a bit to work on. There wasn't enough energy out there on the pitch."
AC Milan striker Ibrahimovic had a chance to put Sweden up 2-0 shortly before halftime after he was taken down in the Moldavian penalty area.
But opposing goalkeeper Stanislav Namasco denied Ibrahimovic's shot, forcing the Swedes to head to the dressing room with a one-goal lead.
The home team failed to create many chances in the second half, having to wait until the 81st minute before Birmingham midfielder Sebastian Larsson gave Sweden a 2-0 lead with his first score wearing a national team jersey.
Larsson's goal ensured Sweden escaped with a win after Moldavia's Alexandru Suvorov netted a score for the visitors in extra time.
The win keeps Sweden in second place in Group E of the Euro 2012 qualifying rounds with 9 points and two games in hand against unbeaten group leader Netherlands, which has 18 points after six matches.
Hungary also has 9 points, but having played six games to Sweden's four, finds itself in third place.