Elmander struck after half-an-hour and again just before the hour in between the opener from Ola Toivonen after 11 minutes and the fourth from substitute Alexander Gerndt in the 88th.
Igor Bugaev got a consolation goal for Moldova in the 61st minute but the home side never threatened to stop the Swedes moving within six points of group leaders Netherlands.
The Dutch are top with 18 points from six matches ahead of Sweden on 15 with a game in hand and Hungary on nine. Sweden next host Finland on Tuesday when Hungary visit San Marino.
Sweden survived an early scare as Alexandru Suvorov clipped the top of the bar with a first-time shot in the sixth minute before going ahead through forward Toivonen.
The striker hit a thunderous swerving shot from distance that keeper Stanislav Namasco could only parry into the net.
Toivonen turned provider on the half-hour, winning the ball in midfield and lifting it into Elmander who fired low across the keeper for his first goal of the campaign.
Elmander then turned home Sebastian Larsson's cross-shot to put the Swedes 3-0 before Bugaev pulled one back, stabbing home from close range to give the noisy home crowd some hope.
NERVY FINISH
Sweden's Andreas Isaksson tipped over a dipping shot from distance by Suvorov in the 67th to preserve their lead.
The winger, who scored against the Swedes in Stockholm in March, had been a constant threat cutting in from the right.
Sensing a nervy finish, Sweden coach Erik Hamren brought on midfielder Pontus Wernbloom for Toivonen in the 69th to close the game out.
A minute later fellow substitute Emir Bajrami could have put the game to bed but fired over with only the keeper to beat.
Hamren's tactic seemed to work as Moldova's attacking threat fell away. Sweden had two more chances in quick succession as the game drew to a close but neither Gerndt nor Olof Mellberg could find a way past Stanislav Namasco.
Gerndt redeemed himself in the 88th when Moldova failed to deal with a long ball, firing home with his left foot to claim his second goal for his country.
