The Canadian world number nine lost 7-6 (7-3) 7-6 (7-5) to the in-form Gulbis on Tuesday, and looked to be heading for another defeat when Haase claimed the opening set 6-4.
However, Raonic hit back by winning the second set by the very same scoreline, before sealing a 10-5 win in the champions tie break.
And it proved to be a day of close encounters, as Philipp Kohlschreiber also needed a decider to see off Italy's Fabio Fognini.
After sharing the opening two sets, the match ended 7-5 4-6 10-4 in Kohlschreiber's favour, although Fognini arguably stole the show with a between-the-legs winner during the second set.
The final match of the day also went the distance, with Kevin Anderson beating Jack Sock 6-4 3-6 10-6.