Short answer
The earliest route that was deliberately laid out, put under the crown’s maintenance-obligation and kept open for wheeled traffic all the way from the Swedish side to the Finnish side was the road that the sources call Norrstigen / Kustlandsvägen and that, after the postal ordinance of 1636, became the Norra postvägen (the Northern Post Road).
It reached the Finnish town of Torneå (Fi. Tornio) – at the head of the Gulf of Bothnia – in the years 1638-1642. From Tornio a similar crown road already existed south-wards into Finland. The crossing of the frontier itself was therefore the short stretch Kalix – Karungi – Torneå, with a royal ferry (and, in winter, an ice-road) over the Torne River; no sea passage was involved. This is the road referred to by contemporaries who, in 1808/09, spoke of “the good new roads into Finland”.

In a little more detail

1. The geographical problem
   Almost the whole Swedish–Finnish boundary is formed by the Gulf of Bothnia. A dry-land contact is only possible at the very top of the gulf, in the Torne (Fi. Tornio) river valley. Any “first over-land” route therefore has to run through that valley.

2. Earlier tracks
   • The Sami, birkarlar and other traders had foot and sledge tracks across the Torne valley at least from the 13th century, but these were not planned, built or maintained roads.
   • A continuous coastal path (Norrstigen) is first mentioned in a Swedish charter of 1328, yet for most of its length it was still only a bridle-track that turned into a winter road on the ice. Under the definition given in the question such tracks do not qualify.

3. Gustav Vasa’s orders (1540s-1550s)
   During the 1540s the Crown began to levy land transport services (landsvägskörslor) from the peasants of Ångermanland, Västerbotten and Österbotten. District bailiffs were told “to clear a road passable for carts as far as Torneå”. Parts were built, but enormous gaps remained and the road could only be used with sledges in winter.

4. The postal ordinance of 1636 and the Northern Post Road
   In 1636 Queen Christina’s regency council issued the general postal ordinance that created a country-wide network of crown post-roads. Every stage was to be rideable or driveable in summer as well as winter, provided with mile-posts and kept in repair by the local farmers under supervision of crown road-masters.

• 1638: post stages laid out from Umeå to Luleå.
• 1639-1640: extension to Kalix.
• 1641-1642: Kalix – Karungi – Torneå finished; ferry chartered for the river-crossing.

The “Vägläggnings-förordning” of 25 May 1642 lists Torneå as the northern terminal of the new land road. Because Finland already had a crown road southwards from Torneå (the older Pohjanmaan maantie), the result was an entirely over-land connection Stockholm – Torneå – Oulu – Turku.

5. Later improvements
   • 1736-1737 the French-Swedish expedition of Maupertuis used the same road for the famous meridian-measurement.
   • 1770s mile-stones (in Finnish as well as Swedish) were erected between Luleå and Torneå.
   • During Gustav III’s war (1788-1790) the road was again widened for artillery pieces; a map in Krigsarkivet, Stockholm (Krigsk. Hk. nr 188) shows its full length.
   These are “the several new good roads into Finland” referred to in the general literature on the 1808/09 war.

6. Why nothing earlier can be called a “road”
   Every source we possess before the 1540s describes the Torne valley connection as a ridstig (“riding path”) or a vinterväg (“winter way”) – practicable on horseback or with sledges, but impassable for wheeled vehicles in summer and never kept up by any public authority. Only after the crown began to impose road-labour and, above all, after the 1636 postal ordinance do we see a deliberately laid-out, permanently maintained, regionally used route – i.e. a real road in the sense defined in the question.

Main sources (easily consulted)
• Erik Lönnroth (ed.) Från Våg till Väg. Trafikverkets historiska skrifter 7 (Stockholm 1998), ch. 4.
• Patrik Bygdén Kustlandsvägen genom Norrbotten (rapport, Vägverket Region Norr, 2006).
• Postmuseum (Stockholm) Norra Postvägen 1636-1912, Meddelanden 24 (1985).
• Risto Oja Postitie ja kruununkyyti Pohjanmaalla 1650-1850 (Oulun Hist. Aikakauskirja 5, 1983).
• Swedish Wikipedia article “Kustlandsvägen (Norrland)” with references to the medieval charters and to the 1642 ordinance.

Therefore – allowing for the river ferry/ice passage – the first genuinely over-land road between Sweden and Finland was the Northern Post Road completed to Tornio in the early 1640s.