Rookie Alyssa Thomas had 24 points and Katie Douglas scored 21 for the Sun (8-10), who have lost four straight.
Tulsa scored the first 12 points and led 19-4 with over a minute left in the first quarter.
''Our starts haven't been good, so we've been playing catch-up going into the second half,'' Sims said. ''We just tried to make sure we had a good start and then build on the start.''
The Shock also outrebounded the Sun 42-27, grabbing 17 offensive boards.
Connecticut rebounded from its slow start and was ahead by a point with about 8 minutes left in the third quarter before Tulsa went ahead for good.
Sims' two free throws gave the Shock a 51-46 advantage with 6:44 remaining in the third. They never trailed again.
''We watched their game at New York (on Tuesday) and New York just took it right at them,'' Sun coach Anne Donovan said. ''(New York) attacked the paint. Put them on their heels right from the jump. And that's what our game plan was to do and unfortunately, that's what they did to us.''
It was the best game of Sims' young career. She was the second overall pick in April's WNBA draft.
"She's been leading up to having a breakout night,'' Tulsa coach Fred Williams said. ''I thought tonight she was able to turn the corner and get a rhythm in her shots. We ran some sets and we did certain things offensively for her and Skylar.''
Sims shot 7 of 10 in the first half for 17 points. She scored eight points during a 10-6 run to close the second quarter and give Tulsa a 43-40 lead.
''My mindset was just to stay aggressive. I don't think that any of us were aggressive last game in New York,'' Sims said of Tuesday's 90-74 loss. ''I just had one of those games. My main focus was to just stay aggressive from the jump and not let up.''
Glory Johnson's three-point play gave the Shock their biggest lead, 88-69, with 4:08 left in the game. Johnson had 14 points and 11 rebounds.
''I said postgame and during the game, we have to do a better job - the five starters - of coming in ready to play,'' Douglas said. ''I don't feel like recently, we've set the tone. We've given teams confidence on our own home court. We've dug ourselves holes.
''Any time you dig yourself a hole in the WNBA, you're going to exert a lot of energy and effort to come back and we're not that good, to be honest, to be digging ourselves a hole against good teams right now. ''