The 22-year-old left-wing, who has played rugby league and American football since being released by Leicester Tigers, showed a devastating turn of pace for both his touchdowns as he took the man-of-the-match award.
The first came after only nine minutes when skipper Lawrence Dallaglio, prop Tim Payne and centre Rob Hoadley were all involved in the close passing move which unleashed Baxter.
The youngster burst through the ineffective tackling of Eion Redden and French international wing David Bory to touch down.
Bath replied with a penalty from Chris Malone and a 15th-minute try from flanker Michael Lipman, who cashed in when a Wasps lineout went wrong.
Hooker Joe Ward´s long throw went straight into the hands of the unmarked Lipman who scored a gift try from close range.
Wasps fly-half Jeremy Staunton kicked two penalties, with Malone replying with his second, as the sides went in level at 11-11 at the end of the first half in which the champions failed to take advantage of their numerical advantage.
Bath had number eight Zak Feaunati sin-binned for a high tackle on home full-back Tom Voyce in the 21st minute.
Malone´s third penalty put Bath 14-11 ahead before Baxter squeezed in at the left corner flag for his second try. Boyce had slipped the ball inside to Hoadley who found the winger with his long pass.
But Malone put Bath back in front with another penalty and a trademark long-range drop goal from 45 metres as Wasps again failed to take advantage of their numerical superiority after Lipman was yellow-carded for killing the ball.
Alex King, who had replaced fly-half Staunton, regained the lead for Wasps with a 70th-minute penalty before Malone missed the chance to turn the tables again with a long-range penalty attempt.
Wasps finally put the issue beyond doubt in the 77th minute.
Following a sustained spell of pressure set up when flanker John Hart did well to charge down an attempted clearance from Malone, Dallaglio sent Voyce in for the home side´s third try from close range and King added the conversion.