Evans, 26, lost the first set 6-3 to 22-year-old wildcard Broady, who is ranked sixth in Britain.
But world number 92 Evans, in his first competitive meeting with Broady, took the second set 6-2 and the final set 8-6 on a tie-break.
The British number four will face Lithuania's Ricardas Berankis, ranked 56 in the world, in the second round.
Dan Evans came from a set down to beat fellow Briton Liam Broady in the first round of the Nottingham Open.