Back came the D-backs, with Tommy Pham's homer immediately restoring their one-run lead, before Ketel Marte extended his postseason hitting streak to 17 games when he drove in Geraldo Perdomo.
That made it 5-3 in the fifth, but the score remained the same until Seager's dramatic ninth-inning shot.
Tantalisingly, the Rangers then twice put the potential winning run on second base before Garcia, named as most valuable player in the AL Championship Series win over the Houston Astros, sent the Texas crowd wild.
While a recent rule change puts a "ghost runner", external on second base at the start of each extra inning during regular-season games, it does not apply in the play-offs - although it ultimately made no difference as Rangers closer Jose Leclerc threw two hitless innings to collect the win in relief.
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