The 10th-seeded Czech, on the hunt for a new coach after splitting with Dani Vallverdu following May's Italian Open, won 7-6(5) 5-7 6-1 7-6(2) -- one of only two matches completed by mid-afternoon as the weather cut a swathe through the programme for the second day running.
"The situation right now, it's fine to me, and (a coach is) definitely something that I'm looking for," he told reporters, adding he wanted a long-term working relationship rather than one focused on individual tournaments.
Berdych, who had led 4-1 in the fourth set when play was called off on Tuesday, served for the match at 5-3 and looked in a desperate hurry to finish things off.
But in damp windy conditions that gradually turned to steady drizzle and forced play to be called off on all outside courts, Dodig broke serve and saved two match points before taking the set to a tiebreak.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
Berdych, runner-up to Rafael Nadal at the All England Club in 2010 and contesting his 52nd consecutive grand slam, steadied his nerves to take the breaker 7-2 when he forced a backhand error from Dodig.
"First of all, I should have been done (with the match) already yesterday," he said. "But... especially when my opponents finished yesterday, I think it's really good that we were able to finish the match."
The Czech, vocal in criticising how French Open officials managed frequent rain interruptions at Roland Garros earlier this month, next plays unseeded German Benjamin Becker.
Rain delays were forecast for the rest of Wednesday and Centre Court action began on schedule at 1200 GMT under the roof, with the most eagerly anticipated clash that of seven-times champion Roger Federer against the tournament's biggest outsider, Britain's world number 772 Marcus Willis.
(reporting by John Stonestreet; editing by Martyn Herman)
