The Sweden midfielder, Celtic's leading scorer with 21 goals in all competitions last season, opened his account for the new campaign when he nodded in Kieran Tierney's cross in the 52nd minute.
Before kick-off, Celtic captain Callum McGregor unfurled the champions flag to celebrate the club's 14th Scottish title in 15 seasons and a fifth consecutive league crown.
Celtic's domestic double under interim boss Martin O'Neill last season convinced the club to retain the 74-year-old on a permanent deal.
But the board's parsimonious approach to the transfer market is a bone of contention for frustrated fans who continue to call for new recruits.
O'Neill handed debuts to Colombian striker Camilo Duran and Kasper Hogh, the Denmark forward who cost a club-record £11 million plus add-ons from Norwegian side Bodo/Glimt.
Celtic pipped Hearts to the title after a thrilling conclusion to last season, but they struggled to get going in the opening 45 minutes of the new campaign.
The breakthrough came shortly after the restart when Tierney delivered an inviting cross from the left and Nygren headed beyond Dundee goalkeeper Owen Goodman.
Celtic pushed for a second goal but were grateful to Viljami Sinisalo for a fine save he made to keep out Dundee substitute Ashley Hay's glancing header.
McGregor struck the post late on as Celtic opened their latest title defence with a victory that could have been more comfortable, though the performance was not entirely convincing.
