Can this be done? Absolutely. It has been done, though not very often: just six teams have accomplished this in a five-game postseason series since the divisional play era began in 1969. Two such teams did it in the era when the LCS itself was five games: the 1982 Brewers and 1984 Padres.
In the early days of the league championship series, home field was not assigned by best record, but instead on a divisional rotation, East getting home field one year, West the next. Thus a team hosted two games, then played three on the road, unlike the 2-2-1 system currently in place.
So for the two teams that failed in that era, they both began with a pair of home wins and lost three straight on the road. In 1984, the team that lost three straight had the better regular-season record; giving home field to the team with the better record didn't seem to occur to MLB's moguls until the wild card era. And even now, it still doesn't apply to the World Series. (Gotta get those hotel rooms, right, Bud?) Would the 1984 Cubs have won with home field? The answer is "probably"; they not only had the best overall record, but best home record in the NL.
When the wild card era began, MLB used the same rotating format it had for home field and the series was in the 2-3 format. That meant the team with the home field "disadvantage" actually could have had more home games if it could sweep the series. This was finally changed to the current 2-2-1 format, with the team with the better record getting the first two games at home, in 1998.
1995: Mariners come back to defeat Yankees
This is the only division series comeback made in the 2-3 format. It seems odd now that a wild card would have had the first two games at home, but the Yankees took advantage by winning both of them, Game 2 on a two-run walkoff homer by Jim Leyritz. Seattle won Game 3 behind Randy Johnson, Game 4 in a slugfest, and then Game 5 also went into extras. Just as he would do six years later for the Diamondbacks in the World Series, Johnson pitched in relief, seemingly losing the game by allowing a 12th inning run. But the Mariners got three straight hits in the last of the 12th, winning it on an Edgar Martinez double that scored Ken Griffey Jr. with the winning run.
1999: Red Sox come back to defeat Indians
As in 1995, the home team, Cleveland, won the first two games. And under the new 2-2-1 format, they knew they'd have at least one more home game if needed. The Red Sox crushed the Indians twice at Fenway Park, 9-3 in Game 3 and 23-7 in Game 4. The latter represents the most runs and most lopsided win ever in a Division Series game. The series returned to Cleveland; both teams' bullpens were shot, but Boston held on for a 12-8 win when Pedro Martinez threw six no-hit innings in relief.
2001: Yankees come back to defeat Athletics
For the first time, a team lost the first two games of a five-game series at home and came back to win the series. The A's took a pair of two-run wins in Yankee Stadium, Game 2 on a shutout by Tim Hudson and Jason Isringhausen. The turning point in Game 3 was the famous Derek Jeter twisting, turning relay to nail Jeremy Giambi at the plate; New York hung on to win 1-0 (there are some who claim catcher Jorge Posada never tagged Giambi). The A's never recovered; they were defeated soundly 9-2 in Game 4 and had to go back to Yankee Stadium for Game 5, which the Yankees won 5-3 -- after spotting Oakland a 2-0 lead.
2003: Red Sox come back to defeat AthleticsThe A's again took a 2-0 series lead, this time at home. Game 3 was tied 1-1 going into the seventh; the A's loaded the bases with one out, but could not score. They withstood furious Red Sox rallies in the eighth and ninth, and the game went into extra innings, where Trot Nixon hit a two-run walkoff homer off Rich Harden in the 11th to win it for Boston. That seemed to crush the A's spirits; Boston came from behind to win Game 4 at Fenway and Game 5 in Oakland to advance.
Since 2003, there's been little drama in these series; just two, Braves/Astros in 2004 and Rangers/Rays in 2010, have even made it to a fifth game. The first of those was tied 1-1 after two games; last year the Rangers nearly blew a 2-0 lead they had fashioned on the road, but Cliff Lee shut down the Rays in Game 5 to allow Texas to move to the ALCS.
So can Arizona come back? The D'backs were 51-30 at home in 2011, while Milwaukee was just 39-42 on the road, the only playoff team this year with a losing road record (though they were much better on the road, 24-13, after the All-Star break). The two teams split a four-game series in Phoenix in July. We'll find out on Tuesday.