This can work
Besides "talk to your DM" your idea is viable. What you and the DM need to work out is "How strong is a gust of wind?"
From the spell descriptions, a good estimate is that it is blowing at least ten mph (though I prefer the nautical knots, the book uses 10 mph which is a bit less). How did I arrive at that?
That much wind, at least 10 mph, will disperse a Fog Cloud. (PHB, p 243 spell description). Fog is similar enough to "dispersing gas or vapor," which the Gust of Wind spell description says it will do. (PHB. p. 249).
The Gust of Wind could be something like a strong breeze, Beaufort scale 25-30 mph, but the book doesn't say that it uses that scale. - idea from nitsua60's comment to this answer - See if your DM will buy that. Since the spell description indicates that it pushes people back, or makes walking into the wind difficult, that larger number is a valid argument.
What does this do to your boat?
If you get at least a 10 MPH tailwind boost that your opponent doesn't have, and a strong wind isn't already blowing, you get just under a quarter of an hour's lead on him (13 minutes per your spell budget), so we figure out now: How far is that?
That depends. How fast can your boat go?
Depending on how granular you DM is with sailing boats and how they work, hull length is the prime factor in maximum possible speed, so you may not get a 10 MPH boost compared to your pursuing boat, but some fraction of that: 1/4 to 2/3.
A conservative and playable estimate would give you a half-mile to one-mile lead, if you are pursued by a boat without your advantage. However, if a strong wind is already blowing (in excess of 25 knots) you may both be at your max speed already, and the extra wind may only make handling your boat more difficult.
If you are both tacking more or less into the wind, at an angle, your tail wind may allow you to increase your speed, depending upon how you direct it ... how many sails does your boat have, and what is its rigging?
Present both to the DM, be happy with Conservative Estimate
You are running with more wind than your pursuit, so getting a lead on your pursuer should be an easy case to make with your DM.
A larger lead might be a tougher sell, depending on how rigorous your DM is about ships and their speeds in real world detail. The other potential inefficiency is that the sail area versus spell area may require a deduction. Per spell description, the gust is 60 feet long and ten feet wide ... that shape may or may not hit 100% of the sail area.