Adams started the decisive rally with a leadoff single off of Daniel Moskos (0-2) and moved to second on a wild pitch. After Joe Mahoney moved Adams to third on a groundout to first base, Avery lined a 1-1 pitch just inside the right field foul line to give the Tides their fourth walk-off win this season.
Pat Neshek (1-0) earned the win with two perfect innings of relief.
Bill Hall got the Tides on the board in the second inning, as he belted an opposite-field home run off Indy starter Rudy Owens. Hall later added an RBI single in the sixth inning as he extended his hitting streak to six games.
Norfolk starter Joel Pineiro was very impressive in his season debut, as he surrendered a run on three hits in six innings of work. Pineiro retired the first 11 batters and faced just two batters over the minimum in the contest, as he struck out five without walking a batter. The 33-year-old veteran threw 55 of his 75 pitches for strikes and went to a three-ball count on just two of the 20 batters he faced.
The only damage the Indians could muster against Pineiro came in the fourth, as Matt Hague reached on a two-out single before scoring on a double by former Tide Jake Fox.
The Indians had knotted the score off of Brad Bergesen in the 7th inning on an RBI double Jeff Clement. Bergesen, who was making his first relief appearance of the 2012 campaign, yielded three hits and three walks while striking out two in three innings of work.
The contest was Norfolk's 5th extra-inning affair this season, including four in 16 games at Harbor Park.
The Tides and Indians wrap up their four-game set on Friday night at Harbor Park, with first pitch slated for 7:15. Steve Johnson (1-1, 2.38) will take the hill for the Tides against left-hander Justin Wilson (2-3, 3.00).