Boston beat Minnesota the final three games of a four-game series, and won for the 11th time in 12 games at Fenway Park.
The Red Sox closed out their longest homestand of the season at 6-5.
Hideki Okajima (1-0) pitched two innings, getting out of a two-on, one-out jam in each, for the win.
Jed Lowrie drew a one-out walk against Jim Hoey (0-1) and Iglesias pinch ran. Crawford lofted one off the wall and Iglesias slid in just ahead of the throw from shortstop Matt Tolbert, knocking catcher Rene Rivera over as he scored.
Minnesota manager Ron Gardenhire was ejected in the ninth inning by home plate umpire Joe West.
The Twins, last in the majors in scoring, have put up three or fewer runs in 10 of their last 12 games, and 22 times this season.
The Twins put runners on first and second with one out in the 10th against Okajima, but Tolbert struck out and Alexi Cassila bounced into a fielder's choice. They did it again in the 11th, but Ben Revere grounded into a fielder's choice and Rivera struck out.
Jacoby Ellsbury singled in the third inning, extending his hitting streak to 18 games for the Red Sox. It's his longest since a career-best 22-gamer in May 2009.
Boston wasted a chance in the bottom of the inning. Dustin Pedroia had a one-out double against Jose Mijares and advanced on Adrian Gonzalez's grounder to second. After Kevin Youkilis was walked intentionally, David Ortiz grounded out into the shift in short right.
The Red Sox broke up the scoreless game in the fifth. Jason Varitek had a leadoff double and advanced to third on Ellsbury's ground out to second. After Pedroia bounced to short with the infield in, pitching coach Rick Anderson visited starter Nick Blackburn before Gonzalez lined the first pitch to left, making it 1-0.
The Twins tied it in the eighth when Jonathan Papelbon blew his first save of the season.
Denard Span reached on an infield hit to third with one out in the eighth against Alfredo Aceves before the right-hander was called for a balk by first base umpire Angel Hernandez. After Tolbert fouled to third, Papelbon entered and Jason Kubel blopped a single over Pedroia's head into right-center.
The late run wasted a solid start by Josh Beckett, who gave up six singles, struck out five and walked one over seven innings.
Minnesota's Nick Blackburn, starting when Francisco Liriano was pushed back a day following a bout with the flu and a sore throat, gave up one run, seven hits, walked three and struck out five over 6 1/3 innings.
The Twins collected singles in each of the second through sixth innings, but stranded a runner each time.
Beckett had his last start cut short after 4 1/3 innings by a two hour and 35-minute rain delay. It was his fourth straight no-decision after winning two straight starts in early April, when he threw 15 innings, allowing just one run.
Notes:
Red Sox manager Terry Francona said before the game he hadn't heard anything from MLB regarding his ejection in Friday's 9-2 loss to the Twins. "I figured if it was that bad I'd probably hear from them," he said. … Revere made a fully-stretched diving catch on Ortiz's liner in the second. … Span robbed Ellsbury with his sliding catch in the seventh.