The tourists, at stumps on Saturday's third day, were 363 for seven, after avoiding the follow-on. But that still left them 190 behind England's first innings 553 for five declared.
They will resume with senior batsman Shivnarine Chanderpaul 63 not out.
The left-hander shared valuable partnerships of 92 with Bravo, whose 56 featured a run-a-ball fifty, and 87 with Denesh Ramdin who made a dashing 60 before falling shortly before the close.
England left-arm spinner Monty Panesar took four for 108 on an unhelpful pitch during a spell of 31 overs in all, including 29 interrupted only by lunch and tea.
West Indies were aided by Harmison and fellow Durham quick Liam Plunkett conceding 16 wides between them - just two shy of the Test innings record of 18 set by West Indies at Old Trafford three years ago.
"It wasn't a surprise for us that he was so bad," Bravo told reporters after Harmison finished the day with figures of none for 95 in 22 overs in what was the 28-year-old bowler's first international match since his disappointing Ashes tour.
"We know he can be world-class but we also know he can spray it around as well," Bravo added.
Plunkett at least had the consolation of taking two important wickets, bowling experienced opener Chris Gayle for 30 and removing Denesh Ramdin, following the wicket-keeper's impressive 60, with the aid of a diving slip catch from Paul Collingwood, the third Durham man in England's XI.
Bravo, who gave his wicket away when he hooked medium-pacer Collingwood to Alastair Cook at deep square leg, insisted West Indies could still avoid defeat in the opening match of this four-Test series.
"Of course we are in the game. Shivnarine Chanderpaul is still there and the other guys need to support him," Bravo said.
Harmison often made life tough for debutant wicket-keeper Matt Prior and his innaccuracy was compounded by England being a bowler short after swing specialist Matthew Hoggard limped off with a thigh injury.
Harmison infamously began England's 5-0 Ashes series thrashing with a huge wide to Justin Langer.
That was the start of a lacklustre trip for Harmison who only took 10 wickets in five Tests in Australia and then announced his retirement from one-day internationals.
Although there wasn't too much in a third day pitch for the quicks, there was arguably even less on offer for Panesar.
"The pitch itself, you are not going to get it to turn from leg to off," the spinner said.
Asked about Harmison, Panesar added: "He wanted to bowl a bit better than he did but he's still positive and he'll be firing tomorrow (Sunday)."
Hoggard, who underwent a scan which revealed a strained adductor muscle, will have his fitness reassessed before play resumes.