In the final the Swiss will play eighth seed Kristof Vliegen of Belgium or Tomas Berdych of the Czech Republic, seeded fifth.
Sweden's Borg, five times Wimbledon champion, set the open era record for consecutive wins on grass - 41 - from 1976 to 1981.
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The triple Wimbledon champion saved a set point at 6-5 in the second set before losing the tiebreak 7-4.
Federer broke early in the final set to build a 3-0 lead. Haas had a chance to break back when Federer was serving for the match but the champion saved the break point before clinching victory in two hours 16 minutes.