Alexeev had two goals, and Vincent Lecavalier, Andreas Karlsson and Brad Richards also scored for Tampa Bay, which has won three of four. The six goals are the most scored by the Lightning this year.
Alexander Ovechkin had a pair of goals, and Mike Green and Brian Sutherby also scored for the Capitals, who had claimed points in five straight games. Washington had been unbeaten (2-0-1) at home.
Playing their first game without injured centre Ryan Craig, the Lightning spread out the offence, a strategy that helped compensate for the absence of their leading goal-scorer. Craig sustained a sprained right knee in Thursday's game against Philadelphia and will miss four weeks.
Trailing 2-1 early in the second period, Tampa Bay took control on goals by St. Louis and Alexeev.
St. Louis had his second goal of the season on a wraparound, scoring at 3:10. Just 22 seconds later, Alexeev put the Lightning up 3-2, whistling a wrist shot from center point past Olie Kolzig.
After failing on three consecutive power plays midway through the second, Tampa Bay went ahead by two goals at 19:01, when Karlsson tapped in the rebound of Dmitry Afanasenkov's shot.
The Lightning took a 1-0 lead 5:51 into the game on a short-handed goal by Lecavalier, who converted a long, lead pass from Nolan Pratt. Ovechkin tied it 48 seconds later on the power play, tapping in a shot from Alexander Semin, who had bulled through three defenders.
At 16:07, Mike Green surprised Tampa Bay goalie Marc Denis with a knuckling drive from the top of the slot, giving the Capitals a 2-1 lead.
Midway through the third period, Alexeev scored his second goal, tipping in Dan Boyle's shot, to make it 5-2.
Sutherby struck with 7:22 left, his first goal of the year, but 29 seconds later, Richards beat Washington's Ben Clymer to a loose puck along the boards and restored Tampa Bay's three-goal lead.
Ovechkin converted on a power play at the left post with 3:57 to go.
Notes: C Alexeev has points in five straight games, a career high. ... RW Boyd Gordon and D Bryan Muir were healthy scratches for the Capitals.