Not an answer to Harveys (yet - I'm running my slow DB query) but I do remember this doubleheader all too well.

-- MWE

OK, answer time:

Bob Welch was, in fact, the last pitcher to pitch a complete game 1-0 shutout in which the only run scored on his own HR.
Pitchers who have done it in the Retrosheet era:

Welch, LAD vs CIN, June 17, 1983 (off Mario Soto, 6th inn)
Juan Pizarro, CHC at NYM, September 16, 1971 (off Tom Seaver, 8th inn)
Jim Bunning, PHI at NYM, May 5, 1965 (off Warren Spahn, 6th inn)
Early Wynn, CHW vs BOS, May 1, 1959 (off Tom Brewer, 8th inn)

The only reliever to do it was Cincinnati's Johnny Klippstein, who homered off Houston's Don McMahon in the 13th inning of a game at Colt Stadium on August 8, 1962. In addition to Perez and Gallardo, Baltimore's Milt Pappas won a 1-0 game with a home run at Memorial Stadium against the Yankees on April 18, 1962. Pappas's HR came off Bill Stafford in the 5th; Hoyt Wilhelm pitched the last three innings of that game.

-- MWE

Further fun facts:

Klippstein's 1962 HR was his fifth, and last, major league HR. He had hit one in each of his first four seasons in the majors (1950-1953), then didn't hit another until that 1962 game.

Pizarro's 1971 HR was also his last major league HR, of 8.

Welch's 1983 HR was his first; he hit just one more, in 1986.

Perez's HR was his first, and so far his only, ML HR.

-- MWE

Gallardo also single-handedly won his start vs. RJ and the Giants.

Gallardo also single-handedly won his start vs. RJ and the Giants.

Not exactly, since it was a 3-run shot to break a 1-1 tie and the Giants scored a run later; Gallardo's HR by itself would not have been enough.

-- MWE